#272660

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	#3482	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

805 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=h,c^2=i,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, 179856);