#480435

order	512
nilpotency class	4
isoclinism	isoclinic class of #480274
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	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄ ◁ C₂×#519 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

964 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318858
maximal normal subgroup of	#72318858

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387631);