#7125201

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	#61189	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1661 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)(3,6)(4,7)(5,8)(9,11)(10,14)(12,15)(13,16)(17,18,23,29)(19,30,33,43)(20,26,38,46)(21,28,22,27)(24,35,42,34)(25,39,44,37)(31,41,36,40)(32,47,48,45);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,19,22,35,23,33,21,34)(18,24,28,43,29,42,27,30)(20,36,39,32,38,31,37,48)(25,41,46,47,44,40,26,45);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,20,23,38)(18,25,29,44)(19,31,33,36)(21,37,22,39)(24,40,42,41)(26,28,46,27)(30,47,43,45)(32,35,48,34);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,20)(18,26)(19,32)(21,39)(22,37)(23,38)(24,41)(25,27)(28,44)(29,46)(30,47)(31,34)(33,48)(35,36)(40,42)(43,45);(2,8)(6,9)(7,16)(14,15)(19,33)(24,42)(30,43)(31,36)(32,48)(34,35)(40,41)(45,47)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=h,d^2=hi,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, 7032397);