#272613

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

905 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)(3,12,9,16)(4,14,15,8)(5,13,7,10)(17,18,23,26,22,28,21,27)(19,29,30,24,20,32,31,25);(1,3)(2,4)(5,14)(6,9)(7,8)(10,16)(11,15)(12,13)(17,19)(18,24)(20,22)(21,31)(23,30)(25,28)(26,32)(27,29);(1,4)(2,8)(3,13)(5,12)(6,15)(7,16)(9,10)(11,14)(17,19,22,20)(18,24,28,25)(21,30,23,31)(26,29,27,32);(1,4)(2,9)(3,11)(5,16)(6,15)(7,12)(8,10)(13,14)(17,20,22,19)(18,25,28,24)(21,30,23,31)(26,29,27,32)

Polycyclic presentation:

2^9|a^2=fi,c^2=i,d^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, 179809);