#495064

order	512
nilpotency class	4
isoclinism	isoclinic class of #33595
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

672 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)(6,11)(7,16)(8,19)(9,21)(12,26)(13,20)(14,24)(15,25)(17,30)(18,32)(22,34)(23,28)(27,33)(29,35);(1,3,4,2)(5,6)(7,12)(8,20)(9,13)(10,15)(11,19)(14,21)(16,28)(17,27)(18,33)(22,24)(23,31)(25,34)(26,32)(29,30)(35,36);(5,7,9,16,10,17,21,30)(6,12,13,28,15,27,14,29)(8,18,19,31,22,32,34,36)(11,23,24,26,25,35,20,33);(7,17)(12,27)(16,30)(18,32)(23,35)(26,33)(28,29)(31,36)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=f,e^2=hi,f^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,h^a=hi,e^b=eh,h^b=hi,d^c=di

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, 402260);