#205942

order	512
nilpotency class	3
isoclinism	isoclinic class of #205765
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	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

975 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)(4,7)(5,15)(6,10)(8,13)(9,12)(14,16)(17,18,21,26,22,25,27,28)(19,23,30,32,29,20,24,31);(1,3,12,14,6,13,5,4)(2,7,15,8,10,16,9,11)(17,19,27,30,22,29,21,24)(18,23,26,31,25,20,28,32);(1,4,5,13,6,14,12,3)(2,8,9,7,10,11,15,16)(17,20,22,23)(18,24,25,30)(19,28,29,26)(21,31,27,32);(3,13)(4,14)(7,16)(8,11)(19,29)(20,23)(24,30)(31,32)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=fi,e^2=hi,f^2=hi,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,d^b=dh,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, 113138);