#362842

order	512
nilpotency class	3
isoclinism	isoclinic class of #24378
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

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

1578 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=eg,d^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,f^c=fi,h^c=hi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 270038);