#214071

order	512
nilpotency class	3
isoclinism	isoclinic class of #184811
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1344 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)(3,7)(4,6)(5,8)(9,10,13,15,12,14,11,16)(17,18)(19,23)(20,28,30,24)(21,25)(22,26)(27,32,29,31);(1,3)(2,5)(4,8)(6,7)(10,14)(15,16)(17,19,22,21)(18,23,26,25)(20,27,30,29)(24,31,28,32);(3,8)(5,7)(17,20,22,30)(18,24,26,28)(19,27,21,29)(23,31,25,32);(2,6)(5,7)(9,11,12,13)(10,15,14,16)(17,21,22,19)(18,25,26,23)(20,29,30,27)(24,32,28,31)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=hi,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cei,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, 121267);