#493040

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	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₂²⋊C₄ ◁ C₂×#519 ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

1212 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)(7,17)(8,13)(9,23)(10,16)(11,14)(12,19)(15,25)(18,31)(20,35)(21,32)(22,26)(24,30)(27,33)(28,34)(29,36);(5,7,10,20,11,21,13,27)(6,12,16,29,14,28,8,18)(9,19,24,31,25,34,26,36)(15,17,22,33,23,32,30,35);(1,3,4,2)(5,8,11,16)(6,13,14,10)(7,18,21,29)(9,22,25,30)(12,27,28,20)(15,24,23,26)(17,31,32,36)(19,33,34,35);(6,14)(8,16)(12,28)(15,23)(17,32)(18,29)(22,30)(33,35)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=fi,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, 400236);