#6628827

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574024
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×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1367 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)(4,8)(5,6)(9,10,14,13)(11,15,12,16)(17,18)(19,23)(20,32)(21,25)(22,27)(24,36)(26,33)(28,43)(29,38)(30,39)(31,41)(34,42)(35,40)(37,44)(45,47)(46,48);(2,6)(7,8)(9,11,14,12)(10,15,13,16)(17,19)(18,23)(20,28)(21,31)(22,29)(24,37)(25,41)(26,39)(27,38)(30,33)(32,43)(34,45)(35,46)(36,44)(40,48)(42,47);(1,3,5,4)(2,7,6,8)(9,10,14,13)(11,16,12,15)(17,20,33,35,22,34,21,36)(18,24,25,42,27,40,26,32)(19,28,30,46,29,45,31,44)(23,37,41,47,38,48,39,43);(1,4,5,3)(2,8,6,7)(9,12,14,11)(10,16,13,15)(19,29)(20,34)(23,38)(24,40)(30,31)(32,42)(35,36)(39,41);(9,13,14,10)(11,15,12,16)(17,21,22,33)(18,25,27,26)(19,30,29,31)(20,35,34,36)(23,39,38,41)(24,32,40,42)(28,44,45,46)(37,47,48,43)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=fghi,d^2=fh,e^2=hi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=ehi,d^c=dhi,e^c=ei,e^d=eh

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6536023);