#273736

order	512
nilpotency class	3
isoclinism	isoclinic class of #273691
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

821 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:

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

Polycyclic presentation:

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

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, 180932);