#362637

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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1332 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=efg,d^2=efi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,h^b=hi,g^c=gi,h^c=hi,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, 269833);