#6467403

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437764
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	#2619	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2250 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)(13,14,17,22,18,21,31,36)(15,23,25,33,26,37,30,19)(16,27,29,43,28,20,34,35)(24,39,41,32,40,38,44,42);(2,3)(13,15,18,26)(14,19,21,33)(16,24,28,40)(17,30,31,25)(20,32,27,42)(22,37,36,23)(29,44,34,41)(35,39,43,38);(1,2,4,3)(6,9)(10,12)(13,16)(14,20)(15,24)(17,29)(18,28)(19,32)(21,27)(22,35)(23,38)(25,41)(26,40)(30,44)(31,34)(33,42)(36,43)(37,39);(1,3,4,2)(14,21)(19,33)(20,27)(22,36)(23,37)(32,42)(35,43)(38,39);(5,7,8,11)(6,10,9,12)(16,28)(20,27)(24,40)(29,34)(32,42)(35,43)(38,39)(41,44)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=h,d^2=h,e^2=gi,f^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=dh,f^b=fi,e^c=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, 6374599);