#6990298

order	512
nilpotency class	3
isoclinism	isoclinic class of #6885129
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1452 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)(17,18,23,28)(19,24,30,39)(20,32,34,44)(21,27,22,26)(25,37,42,35)(29,45,33,48)(31,43,36,40)(38,46,41,47);(1,3,4,8)(2,5,6,7)(9,10,13,15)(11,14,12,16)(17,19)(18,24)(20,33)(21,36)(22,31)(23,30)(25,41)(26,43)(27,40)(28,39)(29,34)(32,48)(35,46)(37,47)(38,42)(44,45);(3,8)(5,7)(9,10)(11,16)(12,14)(13,15)(17,20)(18,25)(19,29)(21,35)(22,37)(23,34)(24,38)(26,32)(27,44)(28,42)(30,33)(31,46)(36,47)(39,41)(40,48)(43,45);(2,6)(5,7)(9,11,13,12)(10,14,15,16)(19,30)(20,34)(24,39)(25,42)(31,36)(32,44)(35,37)(40,43);(9,12,13,11)(10,14,15,16)(17,21,23,22)(18,26,28,27)(19,31,30,36)(20,35,34,37)(24,40,39,43)(25,32,42,44)(29,46,33,47)(38,48,41,45)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6897494);