#150390

order	512
nilpotency class	5
isoclinism	isoclinic class of #150390
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#11744
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂×M₃₂	C₂³
prime radical	#2218	C₂²
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₄×M₃₂ ◁ G
LECS	G ▷ C₂×M₃₂ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

213 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 213 conjugacy classes of subgroups — too many to draw. Only its 62 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2)(3,15,18,48)(4,11)(5,25,28,44)(6,32)(7,34)(8,35)(9,14)(10,17)(12,29,45,30)(13,22)(16,43,33,63)(19,55,20,56)(21,57,36,49)(23,46,27,59)(24,42)(26,41)(31,62,58,60)(37,51)(38,50)(39,61)(40,54)(47,53)(52,64);(1,3)(2,10)(4,20)(5,26)(6,33)(7,16)(8,36)(9,18)(11,41)(12,43)(13,47)(14,40)(15,25)(17,30)(19,24)(21,22)(23,53)(27,61)(28,52)(29,54)(31,50)(32,37)(34,38)(35,39)(42,64)(44,48)(45,63)(46,56)(49,62)(51,58)(55,59)(57,60);(1,4,22,6,9,24,8,7)(2,11,13,32,14,42,35,34)(3,16,36,19,18,33,21,20)(5,27,58,30,28,23,31,29)(10,37,39,41,40,38,47,64)(12,44,46,62,45,25,59,60)(15,43,49,55,48,63,57,56)(17,51,61,26,54,50,53,52)

Polycyclic presentation:

2^9|a^2=e,c^2=fg,d^2=ghi,f^2=h,g^2=h,h^2=i,b^a=bd,d^a=df,f^a=fh,g^a=ghi,h^a=hi,c^b=ch,d^b=dg,e^b=efghi,f^b=fhi,g^b=ghi,h^b=hi,d^c=di,e^d=ei

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286095, #72286130, #72286149, …
maximal normal subgroup of	#72286095, #72286130, #72286149, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57586);