#7074719

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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	#42761	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1542 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=fh,d^2=f,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dfi,e^b=ehi,d^c=dh,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, 6981915);