#493024

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	C₂²×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄ ◁ C₂×#520 ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

1231 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)(7,18)(8,14)(9,15)(10,26)(11,16)(12,17)(13,21)(19,32)(20,29)(22,36)(23,33)(24,30)(25,31)(27,34)(28,35);(5,7,11,22,12,23,8,19)(6,13,16,28,17,29,14,27)(9,20,25,35,10,21,24,34)(15,18,31,32,26,33,30,36);(1,3,4,2)(5,8,12,11)(6,14,17,16)(7,19,23,22)(9,25,10,24)(13,27,29,28)(15,31,26,30)(18,32,33,36)(20,35,21,34);(5,9,12,10)(6,15,17,26)(7,20,23,21)(8,24,11,25)(13,18,29,33)(14,30,16,31)(19,34,22,35)(27,36,28,32)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=fi,d^2=i,e^2=i,g^2=i,b^a=be,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, 400220);