#466518

order	512
nilpotency class	3
isoclinism	isoclinic class of #454375
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

932 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=fi,b^2=efg,c^2=ei,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, 373714);