#481684

order	512
nilpotency class	4
isoclinism	isoclinic class of #481601
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1106 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=hi,d^2=i,e^2=h,b^a=be,c^a=cg,e^a=eg,g^a=gi,d^b=di,e^b=eh,f^b=fgh,f^c=fi,f^e=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 388880);