#387716

order	512
nilpotency class	3
isoclinism	isoclinic class of #213760
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1225 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)(3,5)(4,6)(7,8)(9,10)(11,19,22,15)(12,23)(13,17,26,32)(14,18)(16,25)(20,29,34,27)(21,28)(24,30,37,36)(31,35)(33,39)(38,40);(1,3,5,2)(4,7,8,6)(9,11,14,22)(10,15,18,19)(12,20,25,34)(13,21,26,35)(16,29,23,27)(17,31,32,28)(24,33,37,40)(30,38,36,39);(1,4,5,8)(2,6,3,7)(9,12)(10,16,18,23)(11,20)(13,24)(14,25)(15,27,19,29)(17,30,32,36)(21,33)(22,34)(26,37)(28,38,31,39)(35,40);(2,3)(6,7)(9,13)(10,17,18,32)(11,21)(12,24)(14,26)(15,28,19,31)(16,30,23,36)(20,33)(22,35)(25,37)(27,38,29,39)(34,40)

Polycyclic presentation:

2^9|a^2=h,b^2=egi,c^2=e,d^2=g,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,f^b=fi,h^b=hi,f^c=fi,f^d=fi,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72268304
maximal normal subgroup of	#72268304
maximal quotient of	#72268315

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 294912);