#382389

order	512
nilpotency class	3
isoclinism	isoclinic class of #29634
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#379
prime radical	C₂×#3449	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3572 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)(12,16)(13,18)(14,17)(19,28)(20,26)(21,27)(22,30)(23,29)(24,31)(25,32)(33,38)(34,37)(35,39)(36,40);(2,5)(6,8)(9,11)(10,15)(12,19)(13,21)(14,20)(16,28)(17,26)(18,27)(22,33)(23,32)(24,34)(25,29)(30,38)(31,37)(35,40)(36,39);(1,3)(2,6)(4,7)(5,8)(9,12,13,22,14,23,24,35)(10,16,18,30,17,29,31,39)(11,19,21,33,20,32,34,40)(15,25,27,36,26,28,37,38);(3,7)(6,8)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 289585);