#114703

order	512
nilpotency class	3
isoclinism	isoclinic class of #111405
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	C₂×#121	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

745 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:

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

Polycyclic presentation:

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

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, 21899);