#6633349

order	512
nilpotency class	3
isoclinism	isoclinic class of #6632074
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×#447	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1659 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)(3,5)(4,6)(7,8)(10,14)(11,12)(17,18,23,27)(19,28,31,38)(20,25,32,40)(21,26,22,35)(24,36,39,30)(29,43,44,48)(33,41,34,42)(37,45,47,46);(1,3)(2,5)(4,7)(6,8)(9,10,13,14)(11,16,12,15)(20,32)(25,40)(29,44)(33,34)(37,47)(41,42)(43,48)(45,46);(3,7)(5,8)(9,11,13,12)(10,15,14,16)(17,19,23,31)(18,24,27,39)(20,29,32,44)(21,30,22,36)(25,37,40,47)(26,38,35,28)(33,45,34,46)(41,43,42,48);(3,7)(5,8)(9,12,13,11)(10,15,14,16)(17,20)(18,25)(19,29)(21,34)(22,33)(23,32)(24,37)(26,42)(27,40)(28,43)(30,46)(31,44)(35,41)(36,45)(38,48)(39,47);(2,6)(5,8)(17,21,23,22)(18,26,27,35)(19,30,31,36)(20,33,32,34)(24,38,39,28)(25,41,40,42)(29,45,44,46)(37,43,47,48)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fi,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,g^a=gi,c^b=ch,d^b=dfhi,g^c=gi,e^d=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, 6540545);