#6547136

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

1572 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fi,c^2=i,d^2=h,e^2=hi,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, 6454332);