#382825

order	512
nilpotency class	3
isoclinism	isoclinic class of #213788
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

1176 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)(4,5)(7,8)(9,10,15,11)(12,16,24,19)(13,17,28,20)(14,21,30,18)(22,29,32,25)(23,31,26,33)(27,35,40,34)(36,39,38,37);(9,11,15,10)(12,22,24,32)(13,20,28,17)(14,18,30,21)(16,25,19,29)(23,36,26,38)(27,34,40,35)(31,37,33,39);(2,5)(6,8)(9,12)(10,16)(11,19)(13,26,28,23)(14,29)(15,24)(17,33,20,31)(18,22)(21,32)(25,30)(27,37,40,39)(34,38,35,36);(1,3,4,7)(2,6,5,8)(9,13)(10,17)(11,20)(12,23)(14,27)(15,28)(16,31)(18,34)(19,33)(21,35)(22,36)(24,26)(25,37)(29,39)(30,40)(32,38)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 290021);