#408972

order	512
nilpotency class	3
isoclinism	isoclinic class of #393940
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×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

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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=h,c^2=fi,d^2=efi,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 316168);