#465959

order	512
nilpotency class	3
isoclinism	isoclinic class of #454375
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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

1118 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)(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,20)(17,19)(21,22)(23,27)(24,28)(25,29)(26,32)(30,34)(31,36)(33,35);(6,9)(10,12)(13,15,17,16)(14,18,19,20)(21,23,30,33,25,32,24,31)(22,26,34,36,29,27,28,35);(5,7)(6,10)(8,11)(9,12)(14,19)(18,20)(21,23,25,32)(22,27,29,26)(24,33,30,31)(28,35,34,36);(2,4)(7,11)(10,12)(13,16,17,15)(14,20,19,18)(21,24,25,30)(22,28,29,34)(23,31,32,33)(26,35,27,36)

Polycyclic presentation:

2^9|b^2=fgi,c^2=i,d^2=fi,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, 373155);