#7012236

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1372 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=f,c^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=ehi,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, 6919432);