#6930622

order	512
nilpotency class	3
isoclinism	isoclinic class of #6884969
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	#37379	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1531 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=i,d^2=fh,e^2=i,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,e^c=ei,g^c=gi,e^d=ei

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, 6837818);