#151345

order	512
nilpotency class	5
isoclinism	isoclinic class of #151344
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#12187
commutator, Frattini, cosocle	#381	C₂³
prime radical	#534	D₄
LCS	G ▷ #381 ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂⁴ ◁ #381 ◁ G
LECS	G ▷ #381 ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁

293 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 293 conjugacy classes of subgroups — too many to draw. Only its 36 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=fgh,d^2=f,e^2=i,b^a=bd,c^a=ce,d^a=df,e^a=ei,g^a=gh,d^b=df,e^b=egh,g^b=gi,h^b=hi,d^c=dg,e^c=ehi,f^c=fi,g^c=gi,e^d=ehi,g^d=gi

Conjugacy classes & character table

29 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285485
maximal normal subgroup of	#72285485

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58541);