#7206920

order	512
nilpotency class	3
isoclinism	isoclinic class of #7200022
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

1463 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)(3,5)(4,6)(7,8)(10,14)(11,12)(17,18,23,27)(19,24,22,37)(20,30,33,38)(21,34,36,26)(25,31,41,28)(29,42,35,39)(32,46,47,44)(40,43,48,45);(1,3)(2,5)(4,7)(6,8)(9,10)(11,16)(12,15)(13,14)(17,19,23,22)(18,24,27,37)(20,31,33,28)(21,35,36,29)(25,30,41,38)(26,42,34,39)(32,43,47,45)(40,46,48,44);(3,7)(5,8)(9,11,13,12)(10,15,14,16)(17,20)(18,25)(19,28)(21,32)(22,31)(23,33)(24,38)(26,40)(27,41)(29,43)(30,37)(34,48)(35,45)(36,47)(39,46)(42,44);(2,6)(3,7)(9,11,13,12)(10,16,14,15)(17,21,23,36)(18,26,27,34)(19,29,22,35)(20,32,33,47)(24,39,37,42)(25,40,41,48)(28,43,31,45)(30,44,38,46);(3,7)(5,8)(9,12,13,11)(10,16,14,15)(20,33)(21,36)(25,41)(26,34)(28,31)(29,35)(30,38)(39,42)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=f,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=ehi,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7114116);