#253625

order	512
nilpotency class	3
isoclinism	isoclinic class of #253603
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	C₂×C₄²	C₂⁴
prime radical	#532	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

741 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 160821);