#113324

order	512
nilpotency class	3
isoclinism	isoclinic class of #113321
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

733 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:

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

Polycyclic presentation:

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

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, 20520);