#6492547

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	#2431	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1875 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)(7,11)(10,12)(13,14)(15,24)(16,21)(17,22)(18,20)(19,25)(23,31)(26,36)(27,32)(28,33)(29,37)(30,35)(34,39)(38,42)(40,43)(41,44);(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(13,15)(14,19)(16,28)(17,25)(18,27)(20,34)(21,36)(22,24)(23,32)(26,29)(30,41)(31,39)(33,37)(35,43)(38,44)(40,42);(1,3,4,2)(6,9)(10,12)(14,20)(15,25)(18,31)(19,32)(21,35)(22,23)(24,34)(26,28)(30,42)(33,43)(36,44)(37,38);(6,9)(10,12)(13,16,17,29)(14,21,22,37)(15,26,25,28)(18,30,31,42)(19,33,24,36)(20,35,23,38)(27,40,39,41)(32,43,34,44);(5,7,8,11)(6,10,9,12)(14,22)(16,29)(19,24)(20,23)(26,28)(30,42)(32,34)(40,41)

Polycyclic presentation:

2^9|b^2=fi,c^2=fi,d^2=i,e^2=hi,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, 6399743);