#7503002

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500669
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	#30502	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1844 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)(7,8)(9,11)(10,12)(13,14,17,22)(15,24,26,19)(16,28,29,36)(18,20,31,23)(21,30,37,32)(25,39,40,44)(27,35,42,33)(34,43,38,41);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,15)(14,19)(16,25)(17,26)(18,27)(20,33)(21,34)(22,24)(23,35)(28,44)(29,40)(30,41)(31,42)(32,43)(36,39)(37,38);(2,4)(14,20)(18,31)(19,33)(21,36)(22,23)(24,35)(27,42)(28,37)(30,32)(34,39)(38,44)(41,43);(6,11)(7,12)(13,16,17,29)(14,21,22,37)(15,25,26,40)(18,32,31,30)(19,34,24,38)(20,36,23,28)(27,43,42,41)(33,39,35,44);(5,8,9,10)(6,7,11,12)(15,26)(16,29)(19,24)(21,37)(27,42)(28,36)(30,32)(33,35)

Polycyclic presentation:

2^9|a^2=i,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dfg,g^a=gi,c^b=ch,d^b=dfi,e^b=ei,g^c=gi,e^d=ei,g^d=gi

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, 7410198);