#293264

order	512
nilpotency class	3
isoclinism	isoclinic class of #271131
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	C₂×C₄²	C₂⁴
prime radical	#2856	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

903 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ei,c^2=fi,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,d^b=dh,f^b=fi,g^b=gi,d^c=dhi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 200460);