#113995

order	512
nilpotency class	3
isoclinism	isoclinic class of #113987
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	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

731 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)(35,45)(36,50,54,58)(37,43)(38,42)(39,44)(40,47)(41,53,59,52)(46,55,62,64)(48,56)(49,60)(51,57,63,61);(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)(33,35,39,49)(34,40,44,56)(36,51,54,63)(37,48,38,47)(41,57,59,61)(42,60,43,45)(46,53,62,52)(50,64,58,55);(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)(34,41)(35,46)(37,52)(38,53)(39,54)(40,55)(42,50)(43,58)(44,59)(45,61)(47,51)(48,63)(49,62)(56,64)(57,60)

Polycyclic presentation:

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

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, 21191);