#213319

order	512
nilpotency class	3
isoclinism	isoclinic class of #19671
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₄×#127
prime radical	#3169	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

658 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)(3,9)(4,6)(5,13)(7,12)(8,10)(11,15)(14,16)(17,18,21,27,23,28,22,26)(19,29,20,24,31,32,30,25);(2,6)(7,14)(8,13)(9,15)(17,19,22,30,23,31,21,20)(18,24,27,32,28,25,26,29);(1,3,10,16,4,11,5,12)(2,7,13,15,6,14,8,9)(17,19,21,20,23,31,22,30)(18,25,26,32,28,24,27,29);(3,11)(7,14)(9,15)(12,16)(17,20,21,31,23,30,22,19)(18,24,27,32,28,25,26,29)

Polycyclic presentation:

2^9|a^2=g,b^2=ei,c^2=fi,d^2=g,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=ch,d^c=di

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, 120515);