#254304

order	512
nilpotency class	3
isoclinism	isoclinic class of #254299
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	#2543	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

863 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:

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

Polycyclic presentation:

2^9|b^2=eh,c^2=i,d^2=i,f^2=i,g^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=ghi,d^c=di,e^c=eh,f^c=fi,g^c=gi,f^d=fh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 161500);