#453903

order	512
nilpotency class	3
isoclinism	isoclinic class of #421159
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

611 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=fgh,c^2=gi,d^2=f,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,f^b=fi,h^b=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, 361099);