#493022

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	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₂²⋊C₄ ◁ C₂×#519 ◁ 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)(6,11)(8,18)(12,22)(13,27)(14,24)(16,29)(19,20)(23,28)(25,33)(26,36)(31,32)(34,35);(5,6,9,13,10,14,21,28)(7,12,17,26,15,25,30,35)(8,11,19,23,18,24,20,27)(16,22,31,34,29,33,32,36);(1,3,4,2)(5,7)(6,12)(8,16)(9,17)(10,15)(11,22)(13,26)(14,25)(18,29)(19,31)(20,32)(21,30)(23,34)(24,33)(27,36)(28,35);(7,15)(12,25)(16,29)(17,30)(22,33)(26,35)(31,32)(34,36)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,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, 400218);