#205914

order	512
nilpotency class	3
isoclinism	isoclinic class of #205751
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

921 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)(17,18,21,26,22,25,20,24)(19,23,29,31,28,32,27,30);(1,2,7,13,5,9,4,6)(3,8,14,16,11,15,10,12)(17,19,20,29,22,28,21,27)(18,23,26,30,25,32,24,31);(1,3,4,10,5,11,7,14)(2,8,6,12,9,15,13,16);(2,9)(6,13)(8,15)(12,16)(17,20,22,21)(18,24,25,26)(19,27,28,29)(23,30,32,31)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=f,d^2=i,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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 113110);