#272627

order	512
nilpotency class	3
isoclinism	isoclinic class of #272539
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#18537	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

813 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)(4,8)(5,13)(6,10)(7,11)(9,16)(14,15)(17,18,21,30,23,20,22,26)(19,27,31,25,29,32,28,24);(1,3,7,14)(2,5,11,4)(6,8,16,13)(9,15,10,12)(17,19,23,29)(18,24,20,25)(21,31,22,28)(26,32,30,27);(1,4)(2,8)(3,9)(5,7)(6,15)(10,14)(11,13)(12,16)(17,19,23,29)(18,25,20,24)(21,28,22,31)(26,32,30,27);(1,5)(2,3)(4,7)(6,15)(8,9)(10,13)(11,14)(12,16)(17,20)(18,23)(19,24)(21,26)(22,30)(25,29)(27,31)(28,32)

Polycyclic presentation:

2^9|a^2=fi,b^2=h,c^2=i,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dh,e^b=eh,g^b=ghi,f^c=fi,g^c=ghi,e^d=eh

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, 179823);