#113940

order	512
nilpotency class	3
isoclinism	isoclinic class of #113932
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	#37448	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

1106 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,d^2=i,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fhi,c^b=cf,d^b=di,e^b=eh,g^b=ghi,d^c=di,e^c=ei,g^c=gh

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, 21136);