#253745

order	512
nilpotency class	3
isoclinism	isoclinic class of #253699
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	#3505	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

827 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:

(1,2,6,3)(4,7,12,9)(5,8,13,10)(11,15,16,14)(17,18)(19,20)(21,25)(22,26)(23,27)(24,30)(28,32)(29,31);(1,3,6,2)(4,7)(5,10,13,8)(9,12)(11,14)(15,16)(17,19,23,29)(18,20,27,31)(21,24,22,28)(25,30,26,32);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,15)(13,16)(17,20)(18,24)(19,26)(21,30)(22,32)(23,31)(25,29)(27,28);(1,5)(2,8)(3,10)(4,11)(6,13)(7,14)(9,15)(12,16)(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=ehi,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gh,c^b=cg,g^b=ghi,e^c=eh,f^c=fi,g^c=gi,f^d=fh

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 160941);