#493039

order	512
nilpotency class	4
isoclinism	isoclinic class of #33359
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂²×Q₈	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄ ◁ C₂×#520 ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

1262 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400235);