#188926

order	512
nilpotency class	3
isoclinism	isoclinic class of #188818
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#380
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	C₂×C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

881 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 96122);