#254280

order	512
nilpotency class	3
isoclinism	isoclinic class of #254275
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	#18384	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

889 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)(3,12,14,16)(4,9,6,10)(8,11,15,13)(17,19,24,21)(18,23,26,22)(20,25)(27,30)(28,31)(29,32);(1,3)(2,8)(4,13)(5,14)(6,11)(7,15)(9,16)(10,12)(17,20)(18,25)(19,27)(21,28)(22,31)(23,30)(24,29)(26,32);(1,4,5,6)(2,9,7,10)(3,13,14,11)(8,16,15,12)(17,21,24,19)(18,22,26,23)(20,28,29,27)(25,31,32,30)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 161476);