#7124103

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993875
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	#3492	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1582 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)(17,18,24,28)(19,31,34,46)(20,36,21,27)(22,30,23,29)(25,40,43,35)(26,39,44,38)(32,41,37,42)(33,47,48,45);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,19)(18,25)(20,37)(21,32)(22,40)(23,35)(24,34)(26,42)(27,45)(28,43)(29,46)(30,31)(33,39)(36,47)(38,48)(41,44);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,20)(18,26)(19,32)(21,24)(22,38)(23,39)(25,41)(27,30)(28,44)(29,36)(31,47)(33,40)(34,37)(35,48)(42,43)(45,46);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,21,24,20)(18,27,28,36)(19,33,34,48)(22,38,23,39)(25,42,43,41)(26,29,44,30)(31,47,46,45)(32,35,37,40);(2,8)(6,9)(7,16)(14,15)(18,28)(19,34)(26,44)(27,36)(29,30)(32,37)(33,48)(35,40)

Polycyclic presentation:

2^9|a^2=i,c^2=hi,d^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=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, 7031299);