#354428

order	512
nilpotency class	3
isoclinism	isoclinic class of #3008
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂²×C₄⋊C₄
prime radical	C₂⁵	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1526 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)(3,4)(5,6,8,7)(9,10,14,19,15,20,13,18)(11,21,24,16,25,30,23,31)(12,17,28,34,26,32,27,33)(22,35,38,29,36,40,37,39);(9,11,13,24,15,25,14,23)(10,16,18,30,20,31,19,21)(12,22,27,38,26,36,28,37)(17,29,33,40,32,39,34,35);(2,4)(9,12)(10,17)(11,22)(13,27)(14,28)(15,26)(16,29)(18,33)(19,34)(20,32)(21,35)(23,37)(24,38)(25,36)(30,40)(31,39);(5,7,8,6)(12,26)(17,32)(22,36)(27,28)(29,39)(33,34)(35,40)(37,38)

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=eg,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,e^b=ei,g^b=gi,d^c=di

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261624);