#194945

order	512
nilpotency class	3
isoclinism	isoclinic class of #18414
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1192 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 102141);