#205936

order	512
nilpotency class	3
isoclinism	isoclinic class of #205761
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	C₂×C₄²	C₂⁴
prime radical	C₂×#3207	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1050 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=fh,d^2=i,e^2=hi,f^2=hi,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,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, 113132);