#6832129

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

1475 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)(9,10,12,13)(11,14,15,16)(19,27)(23,37)(28,43)(30,33)(31,38)(39,41)(44,46)(45,48);(1,3)(2,5)(4,8)(6,7)(17,18,21,25)(19,28,27,43)(20,32,34,47)(22,36,29,26)(23,38,37,31)(24,40,42,35)(30,41,33,39)(44,48,46,45);(10,13)(14,16)(17,19,29,38,21,27,22,31)(18,23,36,43,25,37,26,28)(20,33,40,45,34,30,35,48)(24,41,47,44,42,39,32,46);(2,6)(5,7)(20,34)(24,42)(30,33)(32,47)(35,40)(39,41)(44,46)(45,48);(3,8)(5,7)(9,11,12,15)(10,14,13,16)(17,20,29,40,21,34,22,35)(18,24,36,47,25,42,26,32)(19,30,38,48,27,33,31,45)(23,39,43,46,37,41,28,44)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=hi,e^2=gh,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, 6739325);