#444523

order	512
nilpotency class	3
isoclinism	isoclinic class of #417589
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

936 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)(5,6,8,10)(7,11,12,9)(13,14)(15,18)(16,19)(17,20)(21,22,27,32)(23,28,33,35)(24,29,34,36)(25,30,26,31);(5,7,8,12)(6,9,10,11)(13,15,17,16)(14,18,20,19)(21,23)(22,28)(24,25)(26,34)(27,33)(29,30)(31,36)(32,35);(13,15,17,16)(14,19,20,18)(21,24,25,23,27,34,26,33)(22,29,30,28,32,36,31,35);(2,4)(7,12)(9,11)(13,16,17,15)(14,19,20,18)(21,25,27,26)(22,30,32,31)(23,24,33,34)(28,29,35,36)

Polycyclic presentation:

2^9|a^2=ei,b^2=ef,c^2=fgi,d^2=fi,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,g^b=gi,d^c=di

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 351719);