#381427

order	512
nilpotency class	3
isoclinism	isoclinic class of #29610
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#120
prime radical	#23406	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1382 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)(9,10,15,21,14,19,36,20)(11,22,26,33,25,16,27,29)(12,28,32,24,30,37,31,17)(13,18,35,39,34,38,40,23);(2,5)(6,8)(9,11)(10,16)(12,18)(13,24)(14,25)(15,26)(17,34)(19,22)(20,33)(21,29)(23,31)(27,36)(28,40)(30,38)(32,39)(35,37);(1,3)(2,6)(4,7)(5,8)(9,12)(10,17)(11,23)(13,33)(14,30)(15,32)(16,35)(18,26)(19,24)(20,37)(21,28)(22,40)(25,39)(27,38)(29,34)(31,36);(3,7)(6,8)(9,13,15,35,14,34,36,40)(10,18,21,39,19,38,20,23)(11,24,26,37,25,17,27,28)(12,29,32,22,30,33,31,16)

Polycyclic presentation:

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

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, 288623);