#188259

order	512
nilpotency class	3
isoclinism	isoclinic class of #188252
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#3518	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2177 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=h,e^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,e^b=eh,d^c=di,g^c=gh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 95455);