#7202442

order	512
nilpotency class	3
isoclinism	isoclinic class of #7200042
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	#53151	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1762 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=i,d^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=dfhi,e^b=ehi,g^c=gi,e^d=ei

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7109638);