#7193180

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996405
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#24171	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1590 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2,5,9)(3,7,10,6)(4,12,14,8)(11,15,13,16)(17,18,23,28)(19,29,33,43)(20,34,37,42)(21,26,40,48)(22,41,35,27)(24,32,44,31)(25,30,45,36)(38,47,39,46);(1,3,5,10)(2,6,9,7)(4,13,14,11)(8,16,12,15)(17,19)(18,24)(20,21)(22,32)(23,33)(25,26)(27,43)(28,44)(29,41)(30,38)(31,35)(34,47)(36,39)(37,40)(42,46)(45,48);(1,3)(2,7)(4,11)(5,10)(6,9)(8,16)(12,15)(13,14)(17,20,35,30,23,37,22,36)(18,25,41,42,28,45,27,34)(19,21,31,38,33,40,32,39)(24,26,29,46,44,48,43,47);(3,10)(6,7)(11,13)(15,16)(17,21)(18,26)(19,30)(20,32)(22,39)(23,40)(24,42)(25,43)(27,47)(28,48)(29,45)(31,37)(33,36)(34,44)(35,38)(41,46);(1,4,5,14)(2,8,9,12)(3,11,10,13)(6,15,7,16)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=gi,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7100376);