#6989057

order	512
nilpotency class	3
isoclinism	isoclinic class of #6885129
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	#37553	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1524 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,d^2=fhi,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,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, 6896253);