#495865

order	512
nilpotency class	4
isoclinism	isoclinic class of #33607
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#2215	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

732 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,4)(5,6,8,7)(10,15)(12,23)(13,24)(16,25)(17,26)(18,27)(19,30)(21,36)(22,35)(28,29)(31,34)(32,40)(33,39)(37,38);(9,10,16,30,13,19,12,15,14,18,23,29,24,28,25,27)(11,17,31,40,22,32,21,26,20,33,36,38,35,37,34,39);(2,4)(9,11)(10,17)(12,21)(13,22)(14,20)(15,26)(16,31)(18,33)(19,32)(23,36)(24,35)(25,34)(27,39)(28,37)(29,38)(30,40);(5,7,8,6)(10,18)(11,20)(15,27)(19,28)(21,34)(22,35)(29,30)(31,36)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320128, #72323078
maximal normal subgroup of	#72320128, #72323078
maximal quotient of	#72320193

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403061);