#113832

order	512
nilpotency class	3
isoclinism	isoclinic class of #113829
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	#60790	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

851 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=i,e^2=hi,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fhi,c^b=cf,d^b=di,f^b=fh,g^b=ghi,d^c=dhi,e^c=ehi,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, 21028);