#188993

order	512
nilpotency class	3
isoclinism	isoclinic class of #188956
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

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	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1041 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

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, 96189);