#367090

order	512
nilpotency class	3
isoclinism	isoclinic class of #220200
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1444 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=efgh,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,h^c=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, 274286);