#205770

order	512
nilpotency class	3
isoclinism	isoclinic class of #205769
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	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1185 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)(3,8)(4,13)(5,15)(6,10)(7,11)(9,14)(12,16)(17,18)(19,22,26,29)(20,23)(21,24)(25,28,31,32)(27,30);(1,3,5,11,6,12,14,4)(2,7,9,8,10,13,15,16)(17,19)(18,22,24,29)(20,25)(21,26)(23,28,30,32)(27,31);(1,4,14,12,6,11,5,3)(2,8,15,7,10,16,9,13)(17,20,21,27)(18,23,24,30)(19,25,26,31)(22,28,29,32);(20,27)(23,30)(25,31)(28,32)

Polycyclic presentation:

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

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, 112966);