#6467594

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437770
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#25723	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2303 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

(5,6)(7,10)(8,9)(11,12)(13,14,18,23,19,24,30,37)(15,25,28,36,29,40,33,20)(16,21,32,39,31,38,17,22)(26,35,27,41,42,44,43,34);(2,3)(13,15)(14,20)(16,26)(17,27)(18,33)(19,29)(21,34)(22,35)(23,40)(24,36)(25,37)(28,30)(31,42)(32,43)(38,41)(39,44);(1,2,4,3)(6,9)(10,12)(13,16)(14,21)(15,26)(17,30)(18,32)(19,31)(20,34)(22,37)(23,39)(24,38)(25,35)(27,28)(29,42)(33,43)(36,41)(40,44);(1,3,4,2)(13,17,19,32)(14,22,24,39)(15,27,29,43)(16,30,31,18)(20,35,36,44)(21,37,38,23)(25,41,40,34)(26,28,42,33);(5,7,8,11)(6,10,9,12)(16,31)(17,32)(21,38)(22,39)(26,42)(27,43)(34,41)(35,44)

Polycyclic presentation:

2^9|a^2=f,c^2=h,d^2=hi,e^2=g,f^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,e^c=ei,e^d=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6374790);