#369113

order	512
nilpotency class	3
isoclinism	isoclinic class of #196143
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3153	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1842 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)(5,6,11,15,9,14,10,25)(7,16,20,33,18,12,19,28)(8,21,24,30,23,13,26,31)(17,32,22,29,34,27,35,36);(1,2,4,3)(5,7,10,19,9,18,11,20)(6,12,25,33,14,16,15,28)(8,22,26,17,23,35,24,34)(13,29,30,32,21,36,31,27);(5,8,10,24,9,23,11,26)(6,13,25,31,14,21,15,30)(7,17,20,35,18,34,19,22)(12,27,28,29,16,32,33,36);(1,3,4,2)(6,14)(12,16)(13,21)(15,25)(27,32)(28,33)(29,36)(30,31)

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=fg,d^2=ei,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fi,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, 276309);