#205784

order	512
nilpotency class	3
isoclinism	isoclinic class of #205777
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#390
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3191	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1151 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=f,d^2=hi,e^2=hi,f^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fhi,c^b=ch,e^b=ei,g^b=gi,d^c=di

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, 112980);