#494688

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

1988 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)(7,17,21,23)(8,13)(9,24)(10,28,18,15)(11,16)(12,20)(14,27)(19,34,36,32)(22,30)(25,29)(26,31,33,35);(5,7,18,20,11,21,10,13)(6,12,28,23,16,8,15,17)(9,19,33,30,27,36,26,29)(14,25,35,34,24,22,31,32);(2,4)(5,8,11,12)(6,13,16,20)(7,15,21,28)(9,25,27,22)(10,23,18,17)(14,30,24,29)(19,35,36,31)(26,32,33,34);(5,9,11,27)(6,14,16,24)(7,19,21,36)(8,22,12,25)(10,26,18,33)(13,29,20,30)(15,31,28,35)(17,32,23,34)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=i,d^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, 401884);