#6692735

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1425 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)(19,28)(20,32)(21,26)(22,27)(23,25)(24,31)(29,38)(30,43)(33,41)(34,40)(35,42)(36,37)(39,46)(44,48)(45,47);(1,3)(2,5)(4,8)(6,7)(9,11,13,12)(10,14,15,16)(19,29)(20,23)(21,33)(24,37)(25,32)(26,41)(28,38)(31,36)(34,35)(40,42)(44,45)(47,48);(9,12,13,11)(10,14,15,16)(17,19,22,31)(18,24,27,28)(20,29,23,36)(21,30,35,46)(25,38,32,37)(26,39,42,43)(33,44,34,45)(40,48,41,47);(2,6)(5,7)(17,20,22,23)(18,25,27,32)(19,29,31,36)(21,34,35,33)(24,38,28,37)(26,41,42,40)(30,45,46,44)(39,47,43,48);(3,8)(5,7)(17,21,22,35)(18,26,27,42)(19,30,31,46)(20,33,23,34)(24,39,28,43)(25,40,32,41)(29,44,36,45)(37,47,38,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=g,d^2=i,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, 6599931);