#185790

order	512
nilpotency class	3
isoclinism	isoclinic class of #185491
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2246 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)(3,11,14,8)(4,9)(5,7)(6,10)(12,15,13,16)(17,18,21,26,22,24,20,25)(19,27,29,23,30,32,28,31);(2,7)(4,5)(8,15)(9,10)(11,16)(12,13)(17,19,22,30)(18,23,24,31)(20,29,21,28)(25,32,26,27);(1,3)(2,8)(4,12)(5,13)(6,14)(7,15)(9,16)(10,11);(1,4,6,5)(2,7,10,9)(3,12,14,13)(8,15,11,16)(18,24)(23,31)(25,26)(27,32)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,d^2=h,e^2=i,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,d^b=dh,e^b=ei,g^b=ghi,g^c=gh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 92986);