#253422

order	512
nilpotency class	3
isoclinism	isoclinic class of #253411
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	#18328	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

857 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)(3,7)(4,9)(5,13)(6,10)(8,11)(12,16)(14,15)(19,25)(20,28)(22,31)(23,29)(26,32)(27,30);(1,3)(2,7)(4,11)(5,14)(6,12)(8,9)(10,16)(13,15)(17,18,21,24)(19,26)(20,29,28,23)(22,27)(25,31)(30,32);(1,3)(2,8)(4,11)(5,14)(6,12)(7,13)(9,16)(10,15)(17,19)(18,22)(20,30)(21,27)(23,32)(24,26)(25,28)(29,31);(1,4,6,5)(2,9,10,13)(3,11,12,14)(7,8,16,15)(20,28)(23,29)(25,30)(31,32)

Polycyclic presentation:

2^9|b^2=eh,d^2=hi,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=ghi,c^b=cg,f^b=fi,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, 160618);