#504427

order	512
nilpotency class	4
isoclinism	isoclinic class of #499769
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#13336	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

590 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=ei,d^2=fi,e^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,e^b=ei,g^b=gh,h^b=hi,g^c=gh,h^c=hi,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 411623);