#202151

order	512
nilpotency class	3
isoclinism	isoclinic class of #201938
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1051 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=h,e^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fhi,d^b=dh,f^b=fi,d^c=dh,e^c=ei,f^c=fh,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 109347);