#495367

order	512
nilpotency class	4
isoclinism	isoclinic class of #481319
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

822 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)(9,10,15,20,32,24,13,30,14,19,31,23,16,11,21,18)(12,25,28,38,40,22,27,34,26,17,33,35,29,37,36,39);(2,5)(6,7)(9,11,16,10,14,24,32,19)(12,22,29,17,26,37,40,25)(13,30,15,23,21,18,31,20)(27,39,28,38,36,34,33,35);(1,3,4,8)(2,6,5,7)(9,12,14,26)(10,17,19,25)(11,22,24,37)(13,27,21,36)(15,28,31,33)(16,29,32,40)(18,34,30,39)(20,35,23,38);(12,26)(17,25)(22,37)(27,36)(28,33)(29,40)(34,39)(35,38)

Polycyclic presentation:

2^9|a^2=g,b^2=efgh,c^2=f,e^2=h,g^2=hi,h^2=i,b^a=be,c^a=cf,e^a=ei,e^b=eh,g^b=gh,h^b=hi,d^c=di

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, 402563);