#253272

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
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1081 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|b^2=e,d^2=hi,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, 160468);