#362503

order	512
nilpotency class	3
isoclinism	isoclinic class of #190034
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#381
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1314 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=fg,d^2=fi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,e^b=ei,h^b=hi,h^c=hi,e^d=ei,h^d=hi

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, 269699);