#205890

order	512
nilpotency class	3
isoclinism	isoclinic class of #9784
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#390
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#390
prime radical	C₂×#3169	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1085 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)(3,12)(4,8)(5,10)(6,15)(7,11)(9,14)(13,16)(17,18,22,26,21,24,20,25)(19,23,28,30,27,31,29,32);(1,3,6,4,7,14,5,13)(2,8,10,12,11,16,15,9)(17,19,20,28,21,27,22,29)(18,23,26,32,24,31,25,30);(1,4,6,14,7,13,5,3)(2,9,10,8,11,12,15,16)(18,24)(19,27)(25,26)(28,29);(1,5,7,6)(2,10,11,15)(3,13,14,4)(8,12,16,9)(17,20,21,22)(18,25,24,26)(19,28,27,29)(23,30,31,32)

Polycyclic presentation:

2^9|a^2=g,b^2=egi,c^2=fi,d^2=h,e^2=hi,f^2=hi,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,c^b=ch,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, 113086);