#6547134

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535696
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#3038	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1580 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fi,c^2=i,d^2=h,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,g^c=gi,e^d=eh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6454330);