#354928

order	512
nilpotency class	3
isoclinism	isoclinic class of #3057
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

1546 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)(9,10,13,19,15,20,14,18)(11,21,24,16,23,31,25,30)(12,17,27,34,26,32,28,33)(22,35,36,29,38,39,37,40);(5,6,8,7)(9,11,14,25,15,23,13,24)(10,16,19,31,20,30,18,21)(12,22,28,37,26,38,27,36)(17,29,34,39,32,40,33,35);(2,4)(9,12)(10,17)(11,22)(13,27)(14,28)(15,26)(16,29)(18,33)(19,34)(20,32)(21,35)(23,38)(24,36)(25,37)(30,40)(31,39);(5,7,8,6)(11,23)(12,26)(16,30)(17,32)(21,31)(24,25)(27,28)(33,34)

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=eh,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,h^a=hi,d^b=di,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, 262124);