#6633348

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

1673 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)(17,18,23,28)(19,24,33,41)(20,26,35,42)(21,27,22,37)(25,36,43,34)(29,44,30,39)(31,40,32,47)(38,46,48,45);(1,3)(2,5)(4,7)(6,8)(9,10,13,14)(11,16,12,15)(17,19)(18,24)(20,29)(21,31)(22,32)(23,33)(25,38)(26,44)(27,40)(28,41)(30,35)(34,45)(36,46)(37,47)(39,42)(43,48);(3,7)(5,8)(9,11,13,12)(10,15,14,16)(17,20)(18,25)(19,29)(21,34)(22,36)(23,35)(24,38)(26,37)(27,42)(28,43)(30,33)(31,45)(32,46)(39,40)(41,48)(44,47);(3,7)(5,8)(9,12,13,11)(10,15,14,16)(17,20)(18,26)(19,30)(21,36)(22,34)(23,35)(24,39)(25,37)(27,43)(28,42)(29,33)(31,45)(32,46)(38,40)(41,44)(47,48);(2,6)(5,8)(17,21,23,22)(18,27,28,37)(19,31,33,32)(20,34,35,36)(24,40,41,47)(25,42,43,26)(29,45,30,46)(38,39,48,44)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=f,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, 6540544);