#113523

order	512
nilpotency class	3
isoclinism	isoclinic class of #113515
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

760 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:

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

Polycyclic presentation:

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

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, 20719);