#113793

order	512
nilpotency class	3
isoclinism	isoclinic class of #113790
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	#127
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

824 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=i,e^2=hi,f^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fi,c^b=cf,d^b=di,f^b=fh,g^b=ghi,d^c=di,e^c=ehi,f^c=fh,g^c=ghi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 20989);