#191557

order	512
nilpotency class	3
isoclinism	isoclinic class of #19713
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#384
prime radical	C₂⁴	C₄⋊Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

624 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)(3,9)(4,6)(5,13)(7,11)(8,12)(10,15)(14,16)(17,18,22,26,23,27,21,36)(19,28,32,38,30,42,31,24)(20,25,35,40,33,39,34,46)(29,41,44,47,45,48,43,37);(2,6)(7,14)(8,13)(9,15)(17,19,21,31,23,30,22,32)(18,24,26,28,27,38,36,42)(20,29,34,43,33,45,35,44)(25,37,40,41,39,47,46,48);(1,3,5,11,4,10,12,16)(2,7,8,9,6,14,13,15)(17,20)(18,25)(19,29)(21,34)(22,35)(23,33)(24,37)(26,40)(27,39)(28,41)(30,45)(31,43)(32,44)(36,46)(38,47)(42,48);(3,10)(7,14)(9,15)(11,16)(19,30)(20,33)(24,38)(25,39)(28,42)(31,32)(34,35)(40,46)

Polycyclic presentation:

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

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, 98753);