#6376386

order	512
nilpotency class	3
isoclinism	isoclinic class of #60753
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 ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1594 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)(3,10)(4,8)(5,14)(6,9)(7,12)(11,16)(13,15)(17,18,22,26,23,27,21,36)(19,24,31,39,29,38,30,45)(20,32,35,40,33,46,34,25)(28,41,44,47,42,48,43,37);(1,3)(2,7)(4,13)(5,11)(6,12)(8,16)(9,10)(14,15)(17,19,22,31,23,29,21,30)(18,24,26,39,27,38,36,45)(20,28,35,44,33,42,34,43)(25,37,32,41,40,47,46,48);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,15,10,16)(17,20,21,34,23,33,22,35)(18,25,26,32,27,40,36,46)(19,28,30,43,29,42,31,44)(24,37,39,41,38,47,45,48);(1,5,6,4)(2,8,9,14)(3,11,12,13)(7,16,10,15)(19,29)(24,38)(28,42)(30,31)(37,47)(39,45)(41,48)(43,44);(20,33)(25,40)(28,42)(32,46)(34,35)(37,47)(41,48)(43,44)

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,d^b=di,e^c=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6283582);