#363259

order	512
nilpotency class	3
isoclinism	isoclinic class of #188252
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3512	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2450 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:

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

Polycyclic presentation:

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

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, 270455);