#363430

order	512
nilpotency class	3
isoclinism	isoclinic class of #214672
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1216 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)(3,7)(4,5)(6,8)(9,10,15,22,16,20,14,21)(11,23,26,33,12,17,25,18)(13,28,29,36,30,19,31,35)(24,32,38,40,27,37,39,34);(2,5)(6,7)(9,11,16,12)(10,17,20,23)(13,24,30,27)(14,26,15,25)(18,22,33,21)(19,32,28,37)(29,39,31,38)(34,36,40,35);(9,12,16,11)(10,18,20,33)(13,27,30,24)(14,25,15,26)(17,21,23,22)(19,34,28,40)(29,38,31,39)(32,35,37,36);(1,3,4,8)(2,6,5,7)(9,13)(10,19)(11,24)(12,27)(14,29)(15,31)(16,30)(17,32)(18,34)(20,28)(21,35)(22,36)(23,37)(25,38)(26,39)(33,40)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 270626);