#6444112

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2080 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)(7,11)(8,12)(9,10)(13,14,18,22)(15,24,17,20)(16,27,29,43)(19,33,32,23)(21,36,37,30)(25,40,28,42)(26,35,31,39)(34,41,38,44);(1,2)(3,4)(6,10)(11,12)(13,15)(14,20)(16,25)(17,18)(19,26)(21,34)(22,24)(23,35)(27,42)(28,29)(30,41)(31,32)(33,39)(36,44)(37,38)(40,43);(2,4)(5,7,9,8)(6,11,10,12)(13,16,19,30,18,29,32,36)(14,21,23,27,22,37,33,43)(15,25,26,41,17,28,31,44)(20,34,35,42,24,38,39,40);(2,4)(5,8,9,7)(6,12,10,11)(13,17,18,15)(14,20,22,24)(16,28,29,25)(19,31,32,26)(21,34,37,38)(23,35,33,39)(27,42,43,40)(30,44,36,41);(15,17)(16,29)(20,24)(21,37)(26,31)(27,43)(30,36)(35,39)

Polycyclic presentation:

2^9|a^2=i,c^2=fgi,d^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dhi,e^b=ei,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, 6351308);