#392478

order	512
nilpotency class	3
isoclinism	isoclinic class of #32932
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

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

1514 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=f,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,c^b=ci,d^c=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299674);