#113346

order	512
nilpotency class	3
isoclinism	isoclinic class of #113339
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	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

746 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 20542);