#494706

order	512
nilpotency class	4
isoclinism	isoclinic class of #480460
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	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×Q₈ ◁ C₂²×2₋¹⁺⁴ ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1990 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=i,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,h^a=hi,e^b=eh,g^b=gh,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, 401902);