#6470786

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	#37418	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2056 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:

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

Polycyclic presentation:

2^9|b^2=f,c^2=f,d^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,e^b=ei,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, 6377982);