#6829782

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#42883	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1703 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=hi,e^2=gh,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,e^d=ei

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, 6736978);