#113514

order	512
nilpotency class	3
isoclinism	isoclinic class of #113506
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	C₂×#121	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

776 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=i,e^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,f^a=fh,c^b=cf,d^b=di,g^b=ghi,d^c=dh,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, 20710);