#7200061

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	#35757	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2235 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)(3,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,28)(19,24,22,27)(20,31,33,37)(21,26,35,41)(25,36,40,29)(30,38,34,42)(32,44,46,47)(39,45,48,43);(1,3)(2,5)(4,8)(6,7)(9,11)(10,13)(12,15)(14,16)(17,19,23,22)(18,24,28,27)(20,29,33,36)(21,30,35,34)(25,37,40,31)(26,38,41,42)(32,43,46,45)(39,47,48,44);(11,15)(13,16)(17,20)(18,25)(19,29)(21,32)(22,36)(23,33)(24,37)(26,39)(27,31)(28,40)(30,43)(34,45)(35,46)(38,47)(41,48)(42,44);(3,8)(5,7)(10,14)(11,15)(17,21,23,35)(18,26,28,41)(19,30,22,34)(20,32,33,46)(24,38,27,42)(25,39,40,48)(29,43,36,45)(31,44,37,47);(2,6)(5,7)(11,15)(13,16)(17,19,23,22)(18,27,28,24)(20,29,33,36)(21,34,35,30)(25,31,40,37)(26,38,41,42)(32,45,46,43)(39,47,48,44)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=efi,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,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, 7107257);