#7193183

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	#3512	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1551 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)(19,31)(20,37)(21,27)(22,28)(23,29)(24,30)(25,35)(26,38)(32,47)(33,45)(34,42)(36,43)(39,44)(40,46)(41,48);(1,3,5,10)(2,6,9,7)(4,13,14,11)(8,16,12,15)(17,19)(18,25)(20,32)(21,39)(22,34)(23,35)(24,36)(26,27)(28,31)(29,43)(30,42)(33,41)(37,46)(38,40)(44,48)(45,47);(1,3)(2,7)(4,11)(5,10)(6,9)(8,16)(12,15)(13,14)(17,20,23,38,24,39,22,33)(18,26,28,37,30,45,29,44)(19,32,35,40,36,21,34,41)(25,27,31,46,42,47,43,48);(3,10)(6,7)(11,13)(15,16)(17,21)(18,27)(19,33)(20,35)(22,40)(23,41)(24,32)(25,37)(26,43)(28,46)(29,48)(30,47)(31,45)(34,39)(36,38)(42,44);(1,4,5,14)(2,8,9,12)(3,11,10,13)(6,15,7,16)(17,22,24,23)(18,28,30,29)(19,34,36,35)(20,33,39,38)(21,40,32,41)(25,31,42,43)(26,37,45,44)(27,46,47,48)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,e^2=hi,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, 7100379);