#205732

order	512
nilpotency class	3
isoclinism	isoclinic class of #205569
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#390
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#3184	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1040 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)(4,13)(5,7)(6,12)(8,10)(11,15)(14,16)(17,18,22,26,23,19,21,25)(20,28,31,27,29,32,30,24);(1,3,10,16,5,11,4,12)(2,6,13,15,7,14,8,9)(17,19,23,18)(20,27,29,24)(21,25,22,26)(28,30,32,31);(2,7)(6,14)(8,13)(9,15)(17,20,21,30,23,29,22,31)(18,24,26,28,19,27,25,32);(3,11)(6,14)(9,15)(12,16)(20,29)(24,27)(28,32)(30,31)

Polycyclic presentation:

2^9|a^2=g,b^2=eghi,c^2=fh,e^2=hi,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dh,e^b=ei,g^b=gi,d^c=di

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 112928);