#7005585

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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	#45092	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1522 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=fi,c^2=f,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6912781);