#127356

order	512
nilpotency class	3
isoclinism	isoclinic class of #127312
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×H(3,ℤ/4)
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂×#131	C₂³
socle	C₂	#7980
prime radical	C₂×#41	C₂×D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₄×C₈ ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

332 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 124 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)(3,14,17,41)(4,10)(5,24,25,55)(6,28)(7,12)(8,13)(9,16)(11,27,38,56)(15,39,20,42)(18,50,48,63)(19,43,49,61)(21,52,53,58)(22,35)(23,36)(26,34,54,60)(29,37)(30,44)(31,51)(32,46)(33,47)(40,62)(45,57)(59,64);(1,3,6,18,29,49,23,15,8,17,22,48,7,19,4,20)(2,9,10,31,12,32,35,59,13,33,36,30,37,40,28,57)(5,16,26,45,56,62,53,44,25,47,54,64,27,46,21,51)(11,14,34,42,24,43,52,63,38,41,60,39,55,61,58,50);(1,4,7,22,8,23,29,6)(2,10,12,35,13,36,37,28)(3,15,19,18,17,20,49,48)(5,21,27,54,25,53,56,26)(9,30,32,57,33,31,40,59)(11,34,24,52,38,60,55,58)(14,39,43,50,41,42,61,63)(16,44,46,45,47,51,62,64)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=g,d^2=ghi,f^2=gh,g^2=i,b^a=bd,d^a=dg,f^a=fghi,h^a=hi,c^b=ci,e^b=eh,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287291, #72287362, #72287445, …
maximal normal subgroup of	#72287291, #72287362, #72287445, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 34552);