#206008

order	512
nilpotency class	3
isoclinism	isoclinic class of #205799
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×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1002 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)(17,18,20,24,21,23,28,29)(19,22,26,31,25,30,27,32);(1,3,12,14,6,13,5,4)(2,7,15,8,10,16,9,11)(17,19,20,27,21,25,28,26)(18,22,29,31,23,30,24,32);(1,4,5,13,6,14,12,3)(2,8,9,7,10,11,15,16);(3,13)(4,14)(7,16)(8,11)(19,25)(22,30)(26,27)(31,32)

Polycyclic presentation:

2^9|a^2=g,b^2=eghi,c^2=f,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=dhi

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, 113204);