#465653

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

1232 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

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, 372849);