#6376389

order	512
nilpotency class	3
isoclinism	isoclinic class of #60756
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×C₄×D₄
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1579 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=fg,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=cf,e^d=ei

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6283585);