#188173

order	512
nilpotency class	3
isoclinism	isoclinic class of #188062
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2178 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,d^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dhi,g^b=gh,d^c=dh,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 95369);