#113716

order	512
nilpotency class	3
isoclinism	isoclinic class of #113708
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	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#124
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

739 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2,6,10,5,11,16,9)(3,7,13,14,12,4,8,15)(17,18)(19,29)(20,34,38,42)(21,27)(22,26)(23,28)(24,31)(25,37,43,36)(30,39,46,48)(32,40)(33,44)(35,41,47,45);(1,3,5,12)(2,7,11,4)(6,8,16,13)(9,14,10,15)(17,19,23,33)(18,24,28,40)(20,35,38,47)(21,32,22,31)(25,41,43,45)(26,44,27,29)(30,37,46,36)(34,48,42,39);(1,4)(2,8)(3,10)(5,7)(6,15)(9,12)(11,13)(14,16)(17,20,23,38)(18,25,28,43)(19,30,33,46)(21,36,22,37)(24,39,40,48)(26,34,27,42)(29,45,44,41)(31,35,32,47)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,d^2=i,e^2=i,f^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,c^b=cf,d^b=di,f^b=fh,g^b=ghi,d^c=dhi,e^c=ei,f^c=fh,g^c=gi

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, 20912);