#6833747

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806647
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 ◁ #2402 ◁ 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:

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

Polycyclic presentation:

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

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, 6740943);