#253631

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

745 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,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, 160827);