#6803062

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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1582 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fhi,c^2=f,d^2=i,e^2=fhi,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, 6710258);