#6468675

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438017
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1533 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)(3,6)(4,5)(7,8)(10,14)(15,16)(17,18,21,25,22,26,35,42)(19,27,30,36,28,41,34,23)(20,24,33,40,32,39,47,48)(29,43,44,38,31,45,46,37);(9,10,13,14)(11,16,12,15)(17,19,22,28)(18,23,26,36)(20,31,32,29)(21,34,35,30)(24,38,39,37)(25,41,42,27)(33,44,47,46)(40,43,48,45);(2,5)(6,8)(9,11,13,12)(10,15,14,16)(19,28)(23,36)(27,41)(29,31)(30,34)(37,38)(43,45)(44,46);(9,12,13,11)(10,16,14,15)(17,20,22,32)(18,24,26,39)(19,29,28,31)(21,33,35,47)(23,37,36,38)(25,40,42,48)(27,43,41,45)(30,44,34,46);(1,3,4,7)(2,6,5,8)(20,32)(24,39)(29,31)(33,47)(37,38)(40,48)(43,45)(44,46)

Polycyclic presentation:

2^9|a^2=fhi,b^2=h,c^2=hi,d^2=h,e^2=g,f^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,e^d=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6375871);