#7564232

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1743 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)(7,11)(8,12)(9,10)(13,14,18,22)(15,23,27,38)(16,28,30,42)(17,31,25,21)(19,32,34,24)(20,35,36,29)(26,40,41,33)(37,43,44,39);(1,2)(3,4)(6,10)(11,12)(15,24)(16,29)(17,25)(19,23)(20,28)(21,31)(26,41)(27,32)(30,35)(33,40)(34,38)(36,42);(5,7,9,8)(6,11,10,12)(13,15,25,32,18,27,17,24)(14,19,31,38,22,34,21,23)(16,26,35,43,30,41,29,39)(20,33,42,44,36,40,28,37);(2,4)(13,16,25,35,18,30,17,29)(14,20,31,42,22,36,21,28)(15,26,32,43,27,41,24,39)(19,33,38,44,34,40,23,37);(5,8,9,7)(6,11,10,12)(15,27)(19,34)(23,38)(24,32)(26,41)(33,40)(37,44)(39,43)

Polycyclic presentation:

2^9|a^2=i,c^2=fgi,d^2=gi,e^2=f,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=cg,d^b=dh,g^b=gi,h^b=hi,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, 7471428);