#7126604

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 ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1269 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=h,d^2=h,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,e^c=ei,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, 7033800);