#7624350

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624150
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	#62332	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #532 ◁ #2931 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1231 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)(19,27)(21,33)(22,35)(24,38)(25,40)(26,32)(28,44)(29,46)(30,45)(31,47)(34,37)(36,43)(39,48)(41,42);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,18)(19,24)(20,23)(21,25)(22,26)(27,38)(28,30)(29,39)(31,36)(32,35)(33,40)(34,41)(37,42)(43,47)(44,45)(46,48);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,19,23,30)(18,24,20,28)(21,34,36,46)(22,37,35,29)(25,41,31,48)(26,42,32,39)(27,43,45,33)(38,47,44,40);(3,11)(6,9)(10,16)(13,15)(17,20,23,18)(19,28,30,24)(21,31,36,25)(22,32,35,26)(27,44,45,38)(29,42,37,39)(33,47,43,40)(34,48,46,41);(2,8)(6,9)(7,14)(13,15)(18,20)(24,28)(25,31)(26,32)(38,44)(39,42)(40,47)(41,48)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531546);