#6867536

order	512
nilpotency class	3
isoclinism	isoclinic class of #6668044
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	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1455 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)(17,18,21,26,22,25,35,41)(19,23,29,38,28,37,45,48)(20,30,32,40,33,42,34,24)(27,43,44,39,31,47,46,36);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,16)(13,14)(17,19)(18,23)(20,31)(21,29)(22,28)(24,39)(25,37)(26,38)(27,33)(30,47)(32,46)(34,44)(35,45)(36,40)(41,48)(42,43);(10,14)(15,16)(17,20,22,33)(18,24,25,40)(19,27,28,31)(21,34,35,32)(23,36,37,39)(26,42,41,30)(29,46,45,44)(38,47,48,43);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(19,28)(23,37)(27,31)(29,45)(36,39)(38,48)(43,47)(44,46);(3,8)(5,7)(9,12,13,11)(10,15,14,16)(18,25)(19,28)(24,40)(26,41)(27,31)(29,45)(30,42)(44,46)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6774732);