#273359

order	512
nilpotency class	3
isoclinism	isoclinic class of #273307
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#18256	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

959 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=fh,b^2=h,c^2=h,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=ei,f^b=fhi,g^b=gi,e^c=ehi,g^c=gi,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, 180555);