#6877412

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682532
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

1308 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)(17,18,22,26)(19,23,28,37)(20,30,32,47)(21,34,35,25)(24,39,41,33)(27,42,31,46)(29,45,44,38)(36,48,40,43);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,16)(13,14)(17,19,22,28)(18,23,26,37)(20,31,32,27)(21,29,35,44)(24,40,41,36)(25,38,34,45)(30,46,47,42)(33,48,39,43);(2,6)(5,7)(10,14)(15,16)(17,20,21,33,22,32,35,39)(18,24,25,30,26,41,34,47)(19,27,29,43,28,31,44,48)(23,36,38,42,37,40,45,46);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(19,28)(23,37)(27,31)(29,44)(36,40)(38,45)(42,46)(43,48);(3,8)(5,7)(9,12,13,11)(10,15,14,16)(19,28)(20,32)(23,37)(24,41)(29,44)(30,47)(33,39)(38,45)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6784608);