#436680

order	512
nilpotency class	3
isoclinism	isoclinic class of #436680
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15567
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23407	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1165 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)(3,4,13,17)(5,19,22,32)(6,11)(7,10)(8,16,14,18)(9,23,25,28)(12,27)(15,24)(20,30)(21,26)(29,31);(1,3,7,8,6,13,21,14)(2,4,10,16,11,17,26,18)(5,20,23,29,22,12,28,24)(9,15,19,30,25,31,32,27);(1,4)(2,8)(3,10)(5,15)(6,17)(7,18)(9,24)(11,14)(12,19)(13,26)(16,21)(20,32)(22,31)(23,27)(25,29)(28,30);(1,5,21,28,6,22,7,23)(2,9,10,19,11,25,26,32)(3,12,14,29,13,20,8,24)(4,15,16,30,17,31,18,27)

Polycyclic presentation:

2^9|a^2=eg,b^2=f,d^2=ef,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^d=gi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72338468, #72338485, #72338492, …
maximal normal subgroup of	#72338468, #72338485, #72338492, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 343876);