#265178

order	512
nilpotency class	3
isoclinism	isoclinic class of #265163
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3036	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

857 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)(3,10)(4,7)(5,15)(6,9)(8,12)(11,16)(13,14)(17,18,21,25)(19,27,30,23)(20,26,22,24)(28,32,29,31);(1,2,7,15,6,9,5,4)(3,8,12,10,13,16,11,14)(17,19)(18,23)(20,29)(21,30)(22,28)(24,31)(25,27)(26,32);(1,3)(2,8)(4,14)(5,11)(6,13)(7,12)(9,16)(10,15)(18,24)(19,28)(20,22)(25,26)(29,30)(31,32);(1,4,7,2,6,15,5,9)(3,10,12,16,13,14,11,8)(17,20,21,22)(18,24,25,26)(19,29,30,28)(23,31,27,32)

Polycyclic presentation:

2^9|a^2=hi,b^2=ei,d^2=eh,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,d^b=dh,f^b=fhi,g^b=ghi,d^c=di,f^c=fi,g^c=ghi

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, 172374);