#493459

order	512
nilpotency class	4
isoclinism	isoclinic class of #493458
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#1782
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	#62855	C₂
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #3428 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

2257 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319990
maximal normal subgroup of	#72319990
maximal quotient of	#72320021

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400655);