#253876

order	512
nilpotency class	3
isoclinism	isoclinic class of #253843
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	#40657	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

893 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ei,c^2=i,d^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,d^b=dh,f^b=fh,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, 161072);