#6445042

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	#2414	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2027 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)(15,23,26,37)(16,20,28,34)(17,30,24,21)(19,31,33,27)(25,36,39,44)(29,41,38,35)(32,42,43,40);(1,2)(3,4)(6,9)(10,12)(16,28)(20,34)(25,39)(29,38)(32,43)(35,41)(36,44)(40,42);(2,4)(13,15,24,31,18,26,17,27)(14,19,30,37,22,33,21,23)(16,25,38,42,28,39,29,40)(20,32,41,44,34,43,35,36);(2,4)(13,16)(14,20)(15,25)(17,29)(18,28)(19,32)(21,35)(22,34)(23,36)(24,38)(26,39)(27,40)(30,41)(31,42)(33,43)(37,44);(5,7,8,11)(6,10,9,12)(15,26)(16,28)(19,33)(20,34)(23,37)(27,31)(29,38)(35,41)

Polycyclic presentation:

2^9|a^2=i,c^2=gi,e^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dhi,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, 6352238);