#6461280

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438017
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1892 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)(9,10,12,14)(11,13,15,16)(17,18,23,35,22,25,21,27)(19,28,34,24,31,41,30,37)(20,26,36,46,32,39,33,40)(29,42,45,38,43,48,44,47);(2,5)(6,7)(9,11)(10,13)(12,15)(14,16)(17,19,22,31)(18,24,25,37)(20,29,32,43)(21,34,23,30)(26,38,39,47)(27,41,35,28)(33,45,36,44)(40,48,46,42);(1,3,4,8)(2,6,5,7)(10,14)(11,15)(18,25)(24,37)(26,39)(27,35)(28,41)(38,47)(40,46)(42,48);(11,15)(13,16)(17,20)(18,26)(19,29)(21,33)(22,32)(23,36)(24,38)(25,39)(27,40)(28,42)(30,44)(31,43)(34,45)(35,46)(37,47)(41,48);(20,32)(26,39)(29,43)(33,36)(38,47)(40,46)(42,48)(44,45)

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=ghi,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, 6368476);