#7523260

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
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

1774 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=i,d^2=f,e^2=fi,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,e^b=ei,g^b=gi,h^b=hi,e^c=ei,g^c=gi,h^c=hi,g^d=gi,h^d=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7430456);