#6465955

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438997
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1896 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)(13,14,18,22)(15,24,17,20)(16,27,30,42)(19,33,29,23)(21,36,38,31)(25,40,28,43)(26,35,32,39)(34,41,37,44);(2,3)(6,9)(10,12)(13,15)(14,20)(16,28)(17,18)(19,26)(21,37)(22,24)(23,35)(25,30)(27,40)(29,32)(31,44)(33,39)(34,38)(36,41)(42,43);(1,2,4,3)(13,16,29,36,18,30,19,31)(14,21,33,42,22,38,23,27)(15,25,32,44,17,28,26,41)(20,34,39,40,24,37,35,43);(1,3,4,2)(13,17,18,15)(14,20,22,24)(16,25,30,28)(19,32,29,26)(21,37,38,34)(23,35,33,39)(27,40,42,43)(31,41,36,44);(5,7,8,11)(6,10,9,12)(15,17)(16,30)(20,24)(21,38)(26,32)(27,42)(31,36)(35,39)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6373151);