#299827

order	512
nilpotency class	3
isoclinism	isoclinic class of #299805
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#10104	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

863 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)(3,12)(4,13)(5,11)(6,10)(7,9)(8,14)(15,16)(18,24)(19,28)(20,30)(21,22)(25,32)(26,27);(1,3)(2,8)(4,9)(5,13)(6,14)(7,16)(10,12)(11,15)(17,18,21,24,23,27,22,26)(19,29,20,25,30,31,28,32);(1,4,7,12,6,15,5,8)(2,3,11,13,10,14,9,16)(17,19,21,20,23,30,22,28)(18,25,26,29,27,32,24,31);(1,5,6,7)(2,9,10,11)(3,13,14,16)(4,12,15,8)(17,20,21,30,23,28,22,19)(18,25,24,31,27,32,26,29)

Polycyclic presentation:

2^9|b^2=ehi,c^2=fh,d^2=e,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gh,d^c=di,g^c=ghi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 207023);