#113677

order	512
nilpotency class	3
isoclinism	isoclinic class of #113669
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	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

766 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2,5,10,6,9,18,23)(3,11,14,24,13,7,17,20)(4,8,16,21,15,22,28,32)(12,19,27,30,26,25,29,31)(33,34)(35,44)(36,49,51,57)(37,52)(38,42)(39,43)(40,46)(41,50,58,53)(45,54,62,56)(47,55)(48,59)(60,61,64,63);(1,3,6,13)(2,7,9,11)(4,12,15,26)(5,17,18,14)(8,19,22,25)(10,24,23,20)(16,29,28,27)(21,31,32,30)(33,35,39,48)(34,40,43,55)(36,45,51,62)(37,47,38,46)(41,56,58,54)(42,59,52,44)(49,64,57,60)(50,63,53,61);(1,4)(2,8)(3,12)(5,16)(6,15)(7,19)(9,22)(10,21)(11,25)(13,26)(14,27)(17,29)(18,28)(20,30)(23,32)(24,31)(33,36,39,51)(34,41,43,58)(35,45,48,62)(37,50,38,53)(40,54,55,56)(42,57,52,49)(44,60,59,64)(46,61,47,63)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,d^2=i,e^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,f^a=fi,c^b=cf,d^b=di,e^b=eh,g^b=ghi,d^c=dhi,e^c=ei,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, 20873);