#7125184

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993887
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	#3485	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1653 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)(19,30,33,48)(20,34,38,39)(21,27,43,35)(22,28,23,44)(25,31,40,32)(26,37,45,36)(41,46,42,47);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,19,23,32,24,33,22,31)(18,25,28,30,29,40,44,48)(20,35,37,46,38,27,36,47)(21,39,41,45,43,34,42,26);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,20)(18,26)(19,27)(21,40)(22,36)(23,37)(24,38)(25,43)(28,39)(29,45)(30,42)(31,46)(32,47)(33,35)(34,44)(41,48);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,21,24,43)(18,27,29,35)(19,26,33,45)(20,25,38,40)(22,41,23,42)(28,46,44,47)(30,37,48,36)(31,39,32,34);(2,8)(6,9)(7,16)(14,15)(17,22,24,23)(18,28,29,44)(19,31,33,32)(20,36,38,37)(21,41,43,42)(25,30,40,48)(26,39,45,34)(27,46,35,47)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=hi,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=ehi,d^c=dh,g^c=gi,f^d=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7032380);