#253306

order	512
nilpotency class	3
isoclinism	isoclinic class of #253267
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

888 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 160502);