#272576

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

965 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)(4,14)(5,9)(6,16)(7,11)(8,10)(13,15)(17,18,23,26,22,28,21,27)(19,29,30,24,20,32,31,25);(1,3)(2,5)(4,11)(6,14)(7,13)(8,9)(10,15)(12,16)(17,19)(18,24)(20,22)(21,31)(23,30)(25,28)(26,32)(27,29);(1,4,7,5)(2,9,11,14)(3,10,13,16)(6,15,8,12)(17,19,22,20)(18,24,28,25)(21,30,23,31)(26,29,27,32);(1,5,7,4)(2,3,11,13)(6,15,8,12)(9,10,14,16)(17,20,22,19)(18,25,28,24)(21,30,23,31)(26,29,27,32)

Polycyclic presentation:

2^9|a^2=fh,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, 179772);