#7623448

order	512
nilpotency class	4
isoclinism	isoclinic class of #7623311
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
prime radical	#3061	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1223 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fi,g^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gh,h^a=hi,c^b=cf,e^b=ei,g^c=gh,h^c=hi,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530644);