#7007365

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1403 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fi,c^2=i,d^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=ehi,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, 6914561);