#393347

order	512
nilpotency class	3
isoclinism	isoclinic class of #196248
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1528 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)(3,7)(4,5)(6,8)(9,10)(11,19)(12,24,26,17)(13,27)(14,16)(15,22)(18,20)(21,34,37,40)(23,33)(25,31,28,32)(29,35)(30,36,39,38);(2,5)(6,7)(9,11,20,35,14,23,13,22)(10,15,27,33,16,29,18,19)(12,21,25,36,26,37,28,38)(17,30,31,40,24,39,32,34);(1,3,4,8)(2,6,5,7)(10,16)(15,29)(17,24)(18,27)(19,33)(30,39)(31,32)(34,40);(9,12)(10,17)(11,21)(13,28)(14,26)(15,30)(16,24)(18,32)(19,34)(20,25)(22,38)(23,37)(27,31)(29,39)(33,40)(35,36)

Polycyclic presentation:

2^9|a^2=h,b^2=efh,c^2=fi,e^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,g^a=gi,g^b=gi,e^d=ei,h^d=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 300543);