#113066

order	512
nilpotency class	3
isoclinism	isoclinic class of #113058
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	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

789 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^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,e^b=ei,g^b=ghi,d^c=dhi,g^c=gh

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, 20262);