#495188

order	512
nilpotency class	4
isoclinism	isoclinic class of #480476
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

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	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

1274 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=fi,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,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, 402384);