#356964

order	512
nilpotency class	3
isoclinism	isoclinic class of #22455
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1456 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 264160);