#7500708

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500701
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#35659	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2812 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=i,c^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,g^a=gi,c^b=ch,d^b=df,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, 7407904);