#382420

order	512
nilpotency class	3
isoclinism	isoclinic class of #184985
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷×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	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3494 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)(9,10,15,20)(11,16,23,32)(12,24,27,35)(13,19,14,18)(17,28,34,26)(21,37,25,39)(22,36,29,31)(30,40,33,38);(2,5)(6,8)(9,11)(10,16)(12,25)(13,22)(14,29)(15,23)(17,33)(18,31)(19,36)(20,32)(21,27)(24,39)(26,40)(28,38)(30,34)(35,37);(1,3)(2,6)(4,7)(5,8)(9,12,14,28,15,27,13,26)(10,17,19,35,20,34,18,24)(11,21,22,38,23,25,29,40)(16,30,31,37,32,33,36,39);(3,7)(6,8)

Polycyclic presentation:

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

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, 289616);