#205688

order	512
nilpotency class	3
isoclinism	isoclinic class of #205553
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#390
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#3205	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1187 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 112884);