#303912

order	512
nilpotency class	3
isoclinism	isoclinic class of #269211
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#45999	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

960 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)(3,7)(4,6)(5,8)(9,10,12,14)(11,16,15,13)(17,18,22,19)(20,27,30,28)(21,25,29,24)(23,32,26,31);(1,3,5,4)(2,6,8,7)(10,14)(13,16)(17,19,22,18)(20,28,30,27)(21,25,29,24)(23,32,26,31);(1,3,5,4)(2,7,8,6)(9,11,13,10,12,15,16,14)(17,20,29,32,22,30,21,31)(18,23,25,28,19,26,24,27);(1,4,5,3)(2,7,8,6)(10,14)(11,15)(20,30)(23,26)(27,28)(31,32)

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=efh,d^2=ei,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=ghi,d^c=dh,g^c=gh

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 211108);