#6866674

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
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	C₂×#121	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1537 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)(9,10,12,13)(11,14,15,16)(17,18)(19,23)(20,25)(21,24)(22,26)(27,33)(28,36)(29,35)(30,37)(31,40)(32,38)(34,42)(39,44)(41,46)(43,47)(45,48);(1,2)(3,6)(4,8)(5,7)(10,13)(11,15)(17,19,22,28)(18,23,26,36)(20,29,31,41)(21,30,33,42)(24,37,27,34)(25,35,40,46)(32,43,44,48)(38,47,39,45);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,14,13,16)(18,24)(19,28)(21,33)(23,34)(25,38)(26,27)(29,41)(32,44)(35,45)(36,37)(39,40)(46,47);(1,3,5,4)(2,6,7,8)(10,13)(14,16)(17,20,22,31)(18,25,26,40)(19,29,28,41)(21,32,33,44)(23,35,36,46)(24,38,27,39)(30,43,42,48)(34,45,37,47);(1,4,5,3)(2,6,7,8)(18,26)(19,28)(20,31)(24,27)(30,42)(32,44)(35,46)(45,47)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6773870);