#483407

order	512
nilpotency class	4
isoclinism	isoclinic class of #33605
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂²×#107
prime radical	C₂⁴	#107
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

495 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285702, #72285888, #72285908, …
maximal normal subgroup of	#72285702, #72285888, #72285908, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390603);