#463638

order	512
nilpotency class	3
isoclinism	isoclinic class of #454382
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

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

1020 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)(6,9)(7,11)(13,14,17,19)(15,20,16,18)(21,22,26,30)(23,28,32,36)(24,33,31,27)(25,34,35,29);(5,6,8,9)(7,10,11,12)(21,23,25,31,26,32,35,24)(22,27,29,28,30,33,34,36);(5,7,8,11)(6,10,9,12)(13,15,17,16)(14,18,19,20)(23,24)(25,35)(27,36)(28,33)(29,34)(31,32);(2,4)(7,11)(10,12)(13,16,17,15)(14,18,19,20)(21,24,25,23,26,31,35,32)(22,28,29,33,30,36,34,27)

Polycyclic presentation:

2^9|a^2=efg,b^2=e,c^2=f,d^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=di,d^c=de,e^c=ei,g^c=gi,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 370834);