#480431

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	C₂³×D₄	C₂×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

1026 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)(11,16)(12,17)(13,18,26,15)(14,19)(20,31)(21,30)(22,29,35,27)(23,32)(24,33,38,28)(25,34)(36,40,37,39);(10,15)(11,20)(13,26)(16,29)(17,28)(18,19)(21,23)(24,38)(27,31)(30,40)(32,39)(33,34);(2,6)(7,8)(9,11,14,20)(10,16,19,31)(12,23,25,21)(13,22,26,35)(15,27,18,29)(17,32,34,30)(24,37,38,36)(28,40,33,39);(1,3,5,4)(2,7,6,8)(9,12,14,25)(10,17,19,34)(11,21,20,23)(13,24,26,38)(15,28,18,33)(16,30,31,32)(22,36,35,37)(27,39,29,40)

Polycyclic presentation:

2^9|a^2=f,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, 387627);