#7503009

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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1822 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)(15,19,25,35)(16,27,29,39)(17,20,31,22)(21,32,38,30)(24,40,28,44)(26,33,42,36)(34,43,37,41);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,15,18,25)(14,19,23,35)(16,28,29,24)(17,26,31,42)(20,33,22,36)(21,37,38,34)(27,44,39,40)(30,43,32,41);(2,4)(14,20)(17,31)(19,33)(21,27)(22,23)(26,42)(30,32)(34,40)(35,36)(37,44)(38,39)(41,43);(6,11)(7,12)(13,16,18,29)(14,21,23,38)(15,24,25,28)(17,32,31,30)(19,34,35,37)(20,27,22,39)(26,43,42,41)(33,40,36,44);(5,8,9,10)(6,7,11,12)(15,25)(16,29)(19,35)(21,38)(26,42)(27,39)(30,32)(33,36)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=i,e^2=f,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, 7410205);