#392044

order	512
nilpotency class	3
isoclinism	isoclinic class of #32706
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1174 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=ei,b^2=h,c^2=g,g^2=i,b^a=be,c^a=cf,c^b=cg,d^b=def,g^b=gi,d^c=de

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299240);