#253812

order	512
nilpotency class	3
isoclinism	isoclinic class of #253795
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	#25587	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

877 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)(4,12)(5,11)(7,9)(8,16)(10,14)(13,15)(17,18,23,25)(19,21,27,22)(20,28,29,26)(24,31,32,30);(1,3,6,2)(4,13,15,12)(5,11,7,9)(8,14,10,16)(17,19)(18,21)(20,24,29,32)(22,25)(23,27)(26,31,28,30);(1,4)(2,8)(3,10)(5,14)(6,15)(7,16)(9,13)(11,12)(17,20)(18,24)(19,26)(21,31)(22,30)(23,29)(25,32)(27,28);(1,5,6,7)(2,9,3,11)(4,14,15,16)(8,13,10,12)(17,19)(18,21)(20,26)(22,25)(23,27)(24,31)(28,29)(30,32)

Polycyclic presentation:

2^9|a^2=h,b^2=e,d^2=hi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,f^b=fh,g^b=ghi,e^c=eh,f^c=fi,g^c=gi,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, 161008);