#7503046

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	#2540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1819 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)(13,14,18,23)(15,25,16,20)(17,30,31,40)(19,34,33,24)(21,27,37,29)(22,32,39,35)(26,38,28,44)(36,43,41,42);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,15,18,16)(14,20,23,25)(17,26,31,28)(19,27,33,29)(21,24,37,34)(22,36,39,41)(30,44,40,38)(32,42,35,43);(2,4)(13,16,18,15)(14,21,23,37)(17,28,31,26)(19,27,33,29)(20,24,25,34)(22,38,39,44)(30,41,40,36)(32,42,35,43);(6,11)(7,12)(13,17,18,31)(14,22,23,39)(15,26,16,28)(19,35,33,32)(20,36,25,41)(21,38,37,44)(24,30,34,40)(27,43,29,42);(5,8,9,10)(6,7,11,12)(15,16)(20,25)(21,37)(26,28)(27,29)(36,41)(38,44)(42,43)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^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,e^c=ei,g^c=gi,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, 7410242);