#7125196

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993899
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	#61183	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1700 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=h,d^2=h,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,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, 7032392);