#6815752

order	512
nilpotency class	3
isoclinism	isoclinic class of #6814827
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1691 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)(3,10)(4,8)(5,14)(6,9)(7,13)(11,16)(12,15)(17,18)(19,28)(20,25)(21,38)(22,26)(23,27)(24,31)(29,43)(30,44)(32,39)(33,36)(34,47)(35,48)(37,42)(40,45)(41,46);(1,3)(2,7)(4,11)(5,12)(6,13)(8,15)(9,10)(14,16)(17,18)(19,28)(20,33)(21,26)(22,38)(23,27)(24,32)(25,36)(29,43)(30,41)(31,39)(34,40)(35,42)(37,48)(44,46)(45,47);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,15,10,16)(17,19)(18,24,27,39)(20,34)(21,31)(22,32)(23,29)(25,41,42,40)(26,28,38,43)(30,37)(33,47,48,44)(35,46)(36,45);(1,5,6,4)(2,8,9,14)(3,12,13,11)(7,15,10,16)(19,29)(24,39)(28,43)(30,34)(31,32)(40,41)(44,47)(45,46);(3,13)(7,10)(11,12)(15,16)(17,20,21,35,23,37,22,36)(18,25,38,48,27,42,26,33)(19,30,31,45,29,34,32,46)(24,40,43,47,39,41,28,44)

Polycyclic presentation:

2^9|c^2=fgh,d^2=f,e^2=hi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei

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, 6722948);