#7148797

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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1420 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:

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

Polycyclic presentation:

2^9|a^2=fgi,c^2=i,d^2=i,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,c^b=ch,d^b=dfi,e^b=eh,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, 7055993);