#6692729

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1448 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)(9,10,13,15)(11,14,12,16)(17,18)(19,27)(20,25)(21,24)(22,26)(23,29)(28,37)(30,45)(31,43)(32,48)(33,39)(34,42)(35,40)(36,44)(38,47)(41,46);(1,3)(2,5)(4,8)(6,7)(9,11,13,12)(10,14,15,16)(17,19,21,29)(18,23,24,27)(20,32,33,46)(22,28,35,31)(25,41,39,48)(26,43,40,37)(30,34,47,36)(38,42,45,44);(9,12,13,11)(10,14,15,16)(18,24)(19,28)(22,35)(23,37)(25,39)(27,43)(29,31)(30,46)(32,47)(34,36)(38,41)(45,48);(2,6)(5,7)(19,29)(20,33)(23,27)(25,39)(28,31)(34,36)(37,43)(42,44);(3,8)(5,7)(17,20,21,33)(18,25,24,39)(19,30,29,47)(22,36,35,34)(23,38,27,45)(26,44,40,42)(28,46,31,32)(37,41,43,48)

Polycyclic presentation:

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

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, 6599925);