#259848

order	512
nilpotency class	3
isoclinism	isoclinic class of #245999
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#447	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1155 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)(4,5)(6,8)(9,10,12,14)(11,13,15,16)(17,18)(19,27)(20,25)(21,24)(22,26)(23,29)(28,32)(30,31);(2,5)(6,7)(9,11)(10,13)(12,15)(14,16)(17,19,20,29)(18,23,25,27)(21,30,22,28)(24,32,26,31);(13,16)(18,24)(19,28)(21,22)(25,26)(29,30)(31,32);(1,3,4,8)(2,6,5,7)(18,25)(23,27)(24,26)(31,32)

Polycyclic presentation:

2^9|a^2=i,b^2=h,d^2=eh,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=gh,f^c=fh,g^c=gh

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 167044);