#7005693

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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	#42806	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1584 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)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,21,25,22,26,34,40)(19,27,29,36,30,39,33,23)(20,24,32,38,31,37,44,47)(28,41,43,46,42,48,45,35);(2,5)(3,7)(9,10)(11,15)(12,14)(13,16);(9,11)(10,13)(12,16)(14,15)(17,19,22,30)(18,23,26,36)(20,28,31,42)(21,33,34,29)(24,35,37,46)(25,39,40,27)(32,45,44,43)(38,48,47,41);(1,3,4,7)(2,6,5,8)(11,16)(13,15)(17,20,22,31)(18,24,26,37)(19,28,30,42)(21,32,34,44)(23,35,36,46)(25,38,40,47)(27,41,39,48)(29,43,33,45);(10,14)(13,15)(20,31)(24,37)(28,42)(32,44)(35,46)(38,47)(41,48)(43,45)

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,d^c=dh,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, 6912889);