#190509

order	512
nilpotency class	3
isoclinism	isoclinic class of #184811
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1400 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=ei,g^2=i,b^a=be,c^a=cf,f^a=fh,d^c=di,g^c=gh

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, 97705);