#392258

order	512
nilpotency class	3
isoclinism	isoclinic class of #184979
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

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	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3454 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)(11,15)(12,23,25,16)(13,19)(14,18)(17,24,32,26)(20,34,36,28)(21,31)(22,30)(27,33)(29,35,39,37)(38,40);(2,5)(6,8)(9,11,14,22)(10,15,18,30)(12,20,25,36)(13,21,27,38)(16,28,23,34)(17,29,32,39)(19,31,33,40)(24,35,26,37);(1,3,4,7)(2,6,5,8)(9,12)(10,16)(11,20)(13,24)(14,25)(15,28)(17,19)(18,23)(21,35)(22,36)(26,27)(29,31)(30,34)(32,33)(37,38)(39,40);(9,12)(10,17)(11,20)(13,26)(14,25)(15,29)(16,19)(18,32)(21,37)(22,36)(23,33)(24,27)(28,31)(30,39)(34,40)(35,38)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=e,g^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fi,h^c=hi,g^d=gi,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, 299454);