#362091

order	512
nilpotency class	3
isoclinism	isoclinic class of #220190
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1270 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=f,d^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,h^b=hi,g^c=gi,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, 269287);