#7562297

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518964
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	#45093	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2214 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)(8,12)(9,10)(13,14,19,24)(15,25,28,35)(16,30,34,38)(17,23,18,22)(20,26,37,29)(21,31,40,33)(27,39,32,36)(41,42,44,43);(1,2)(3,4)(6,10)(11,12)(15,26)(16,31)(17,18)(20,35)(21,38)(22,23)(25,37)(27,32)(28,29)(30,40)(33,34)(36,39);(5,7,9,8)(6,11,10,12)(13,15,17,29,19,28,18,26)(14,20,22,25,24,37,23,35)(16,32,33,42,34,27,31,43)(21,39,30,44,40,36,38,41);(2,4)(5,8,9,7)(6,12,10,11)(13,16,18,31,19,34,17,33)(14,21,23,38,24,40,22,30)(15,27,26,42,28,32,29,43)(20,36,35,44,37,39,25,41);(6,10)(11,12)(15,28)(20,37)(25,35)(26,29)(27,32)(36,39)(41,44)(42,43)

Polycyclic presentation:

2^9|a^2=i,c^2=fg,d^2=fgi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cgi,d^b=dh,g^b=gi,h^b=hi,d^c=di,e^c=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, 7469493);