#466120

order	512
nilpotency class	3
isoclinism	isoclinic class of #454375
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

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

1270 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:

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

Polycyclic presentation:

2^9|b^2=efg,c^2=i,d^2=ef,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de,g^c=gi

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, 373316);