#6820656

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806671
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	#2619	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1557 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)(3,6)(4,5)(7,8)(9,10,13,15)(11,14,12,16)(21,35)(25,41)(31,45)(34,42)(36,40)(37,46)(43,48)(44,47);(2,5)(3,7)(9,11,13,12)(10,14,15,16)(17,18,22,27)(19,28,33,39)(20,24,29,26)(21,36,35,40)(23,30,38,32)(25,42,41,34)(31,46,45,37)(43,44,48,47);(9,12,13,11)(10,14,15,16)(17,19,29,30,22,33,20,32)(18,23,24,39,27,38,26,28)(21,31,42,44,35,45,34,47)(25,37,40,48,41,46,36,43);(2,5)(6,8)(17,20,22,29)(18,24,27,26)(19,30,33,32)(21,34,35,42)(23,28,38,39)(25,40,41,36)(31,44,45,47)(37,43,46,48);(1,3,4,7)(2,6,5,8)(17,21,29,42,22,35,20,34)(18,25,24,40,27,41,26,36)(19,31,30,44,33,45,32,47)(23,37,39,48,38,46,28,43)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=ghi,d^2=i,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6727852);