#6990112

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907869
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	#1999	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1373 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)(3,6)(4,8)(5,7)(9,10,12,14)(11,15,16,13)(17,18)(19,27)(20,31)(21,25)(22,35)(23,29)(24,33)(26,34)(28,41)(30,37)(32,46)(36,48)(38,47)(39,43)(40,44)(42,45);(1,2,6,3)(4,7,5,8)(9,11,12,16)(10,13,14,15)(17,19,21,29)(18,23,25,27)(20,28,32,42)(22,30,34,44)(24,36,38,39)(26,37,35,40)(31,45,46,41)(33,43,47,48);(1,3,6,2)(4,7,5,8)(11,16)(13,15)(17,20,22,33,21,32,34,47)(18,24,26,31,25,38,35,46)(19,28,30,43,29,42,44,48)(23,36,37,45,27,39,40,41);(1,4,6,5)(2,7,3,8)(10,14)(13,15)(18,25)(19,29)(20,32)(26,35)(30,44)(33,47)(36,39)(41,45);(1,5,6,4)(2,7,3,8)(19,29)(23,27)(28,42)(30,44)(36,39)(37,40)(41,45)(43,48)

Polycyclic presentation:

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

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, 6897308);