#7206161

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	#31066	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1572 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)(3,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,29)(19,31,35,25)(20,37,21,27)(22,30,24,28)(26,39,44,38)(32,41,33,46)(34,42,36,43)(40,47,45,48);(1,3,6,2)(4,7,5,8)(9,11)(10,13)(12,15)(14,16)(17,19,23,35)(18,25,29,31)(20,32,21,33)(22,34,24,36)(26,40,44,45)(27,41,37,46)(28,42,30,43)(38,47,39,48);(11,15)(13,16)(17,20,23,21)(18,26,29,44)(19,32,35,33)(22,38,24,39)(25,40,31,45)(27,28,37,30)(34,47,36,48)(41,42,46,43);(1,4,6,5)(2,7,3,8)(10,14)(11,15)(17,21,23,20)(18,27,29,37)(19,33,35,32)(22,38,24,39)(25,41,31,46)(26,28,44,30)(34,47,36,48)(40,42,45,43);(1,5,6,4)(2,7,3,8)(11,15)(13,16)(17,22,23,24)(18,28,29,30)(19,34,35,36)(20,38,21,39)(25,42,31,43)(26,37,44,27)(32,47,33,48)(40,46,45,41)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,d^2=f,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=dfhi,e^b=eh,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, 7113357);