#272592

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
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

935 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=h,c^2=h,d^2=h,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, 179788);