#272546

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	#62136	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

800 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)(3,11)(4,7)(5,9)(6,10)(8,16)(12,13)(14,15)(17,18,23,25,22,20,21,31)(19,26,30,27,29,24,28,32);(1,3)(2,4)(5,15)(6,12)(7,16)(8,13)(9,11)(10,14)(17,19)(18,24)(20,26)(21,28)(22,29)(23,30)(25,32)(27,31);(1,4,6,14)(2,7,10,15)(3,8,12,9)(5,11,16,13)(17,20)(18,22)(19,27)(21,31)(23,25)(24,28)(26,30)(29,32);(1,4,6,14)(2,3,10,12)(5,13,16,11)(7,8,15,9)(17,21,22,23)(18,25,20,31)(19,28,29,30)(24,32,26,27)

Polycyclic presentation:

2^9|a^2=f,c^2=h,d^2=hi,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, 179742);