#6814025

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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1305 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=g,c^2=hi,d^2=f,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=ei,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, 6721221);