#7193178

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	#3511	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1569 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)(19,29,33,41)(20,34,35,48)(21,26,38,47)(22,39,40,27)(24,32,42,31)(25,43,44,30)(36,46,37,45);(1,3,5,10)(2,6,9,7)(4,13,14,11)(8,16,12,15)(17,19,23,33)(18,24,28,42)(20,21,35,38)(22,32,40,31)(25,26,44,47)(27,41,39,29)(30,37,43,36)(34,46,48,45);(1,3)(2,7)(4,11)(5,10)(6,9)(8,16)(12,15)(13,14)(17,20,22,30,23,35,40,43)(18,25,27,34,28,44,39,48)(19,21,32,37,33,38,31,36)(24,26,41,46,42,47,29,45);(3,10)(6,7)(11,13)(15,16)(17,21)(18,26)(19,30)(20,31)(22,37)(23,38)(24,34)(25,29)(27,46)(28,47)(32,35)(33,43)(36,40)(39,45)(41,44)(42,48);(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=hi,c^2=g,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, 7100374);