#360315

order	512
nilpotency class	3
isoclinism	isoclinic class of #24289
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	D₄×C₂²⋊C₄
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1883 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)(7,16)(8,17,23,30)(9,24)(10,14)(11,15)(12,22,19,21)(13,25)(18,27)(20,28,34,32)(26,29)(31,35,36,33);(1,2)(3,4)(5,7,10,6,11,14,16,15)(8,19,22,30,23,12,21,17)(9,18,26,13,27,24,25,29)(20,32,33,36,34,28,35,31);(2,4)(5,8)(6,12)(7,17)(9,20)(10,21)(11,23)(13,28)(14,30)(15,19)(16,22)(18,31)(24,36)(25,33)(26,35)(27,34)(29,32);(5,9)(6,13)(7,18)(8,20)(10,26)(11,27)(12,28)(14,24)(15,29)(16,25)(17,31)(19,32)(21,35)(22,33)(23,34)(30,36)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 267511);