#362969

order	512
nilpotency class	3
isoclinism	isoclinic class of #189674
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#380
prime radical	C₂⁴	C₂²×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#423 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

998 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eg,c^2=g,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,d^b=di,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 270165);