#6557581

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535815
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

1295 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=gi,c^2=fi,d^2=fgi,e^2=fg,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=ei,d^c=di,e^d=eh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6464777);