#6487738

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470185
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	#2619	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1964 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6394934);