#254317

order	512
nilpotency class	3
isoclinism	isoclinic class of #254299
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#24140	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1054 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=ei,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=ghi,c^b=cg,f^b=fi,g^b=ghi,d^c=di,e^c=eh,f^c=fi,g^c=gi,f^d=fh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 161513);