#438014

order	512
nilpotency class	3
isoclinism	isoclinic class of #417581
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 ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1366 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 345210);