#7202425

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	#35737	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1767 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)(17,18,22,28)(19,30,33,24)(20,26,35,42)(21,29,23,27)(25,37,41,34)(31,45,47,39)(32,40,36,43)(38,48,44,46);(2,5)(3,7)(9,11)(10,13)(12,15)(14,16)(17,19,22,33)(18,24,28,30)(20,31,35,47)(21,36,23,32)(25,38,41,44)(26,39,42,45)(27,43,29,40)(34,48,37,46);(11,15)(13,16)(17,20)(18,25)(19,31)(21,34)(22,35)(23,37)(24,38)(26,27)(28,41)(29,42)(30,44)(32,46)(33,47)(36,48)(39,43)(40,45);(1,3,4,7)(2,6,5,8)(10,14)(11,15)(17,20)(18,26)(19,31)(21,37)(22,35)(23,34)(24,39)(25,27)(28,42)(29,41)(30,45)(32,48)(33,47)(36,46)(38,43)(40,44);(2,5)(6,8)(11,15)(13,16)(17,21,22,23)(18,27,28,29)(19,32,33,36)(20,34,35,37)(24,40,30,43)(25,26,41,42)(31,46,47,48)(38,45,44,39)

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=fi,e^2=i,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, 7109621);