#6484925

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470374
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

1431 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)(3,7)(4,5)(6,8)(9,10)(11,16)(12,15)(13,14)(17,18)(19,27)(20,32)(21,36)(22,25)(23,31)(24,34)(26,28)(29,45)(30,42)(33,41)(35,40)(37,48)(38,43)(39,46)(44,47);(2,5)(6,7)(10,14)(11,12)(17,19,28,43,22,30,21,31)(18,23,36,42,25,38,26,27)(20,33,44,46,34,29,35,48)(24,39,47,41,32,37,40,45);(1,3,4,8)(2,6,5,7)(9,11,13,12)(10,15,14,16)(17,20)(18,24)(19,29)(21,35)(22,34)(23,37)(25,32)(26,40)(27,41)(28,44)(30,33)(31,46)(36,47)(38,39)(42,45)(43,48);(9,12,13,11)(10,15,14,16)(19,30)(23,38)(27,42)(29,33)(31,43)(37,39)(41,45)(46,48);(10,14)(15,16)(18,25)(20,34)(23,38)(26,36)(27,42)(29,33)(35,44)(46,48)

Polycyclic presentation:

2^9|b^2=fg,c^2=gi,d^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,e^c=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, 6392121);