#481799

order	512
nilpotency class	4
isoclinism	isoclinic class of #481619
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄⋊C₄	C₂⁴
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄ ◁ C₂×#518 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

949 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)(9,10,13,18,14,17,27,32)(11,19,21,33,22,15,28,30)(12,23,25,34,24,16,26,31)(20,29,37,39,36,35,38,40);(1,2)(3,6)(4,8)(5,7)(9,11,14,22)(10,15,17,19)(12,20,24,36)(13,21,27,28)(16,29,23,35)(18,33,32,30)(25,37,26,38)(31,40,34,39);(9,12)(10,16)(11,20)(13,26)(14,24)(15,29)(17,23)(18,34)(19,35)(21,38)(22,36)(25,27)(28,37)(30,40)(31,32)(33,39);(1,3,5,4)(2,6,7,8)(12,24)(16,23)(20,36)(25,26)(29,35)(31,34)(37,38)(39,40)

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=h,e^2=h,f^2=i,b^a=be,c^a=cg,e^a=eg,g^a=gi,e^b=eh,f^b=fgh,d^c=di,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, 388995);