#494741

order	512
nilpotency class	4
isoclinism	isoclinic class of #480482
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×Q₈	C₂⁴
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×D₄ ◁ C₂²×2₋¹⁺⁴ ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

1978 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egi,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,h^a=hi,e^b=ehi,g^b=ghi,h^b=hi,d^c=di,g^e=gi

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, 401937);