#255133

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	#37752	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1281 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,21,25)(19,26,28,31)(20,30,22,24)(23,27,32,29);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,18)(19,26)(20,24)(21,25)(22,30)(23,27)(28,31)(29,32);(1,3,4,7)(2,6,5,8)(13,16)(17,19,21,28)(18,23,25,32)(20,27,22,29)(24,31,30,26);(2,5)(6,8)(17,20,21,22)(18,24,25,30)(19,27,28,29)(23,31,32,26)

Polycyclic presentation:

2^9|a^2=h,c^2=e,d^2=h,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, 162329);