#381392

order	512
nilpotency class	3
isoclinism	isoclinic class of #29602
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₄³

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 ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1100 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)(15,18)(16,17)(19,20)(23,29)(24,31)(26,32)(28,34)(30,35)(33,36);(6,9)(10,12)(14,17)(15,19)(21,22)(23,26)(24,28)(25,27)(29,32)(30,33)(31,34)(35,36);(7,11)(10,12)(13,15)(14,18)(16,19)(17,20)(21,23,24,29,25,30,31,35)(22,26,28,32,27,33,34,36);(5,7,8,11)(6,10,9,12)(15,19)(18,20)(22,27)(26,33)(28,34)(32,36)

Polycyclic presentation:

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

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, 288588);