#117310

order	512
nilpotency class	3
isoclinism	isoclinic class of #117305
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#126
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

497 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)(3,12)(4,16,20,28)(5,21)(6,10)(7,11)(8,14)(9,19,27,18)(13,26,17,23)(15,24)(22,25)(29,30,32,31)(33,34,38,44,39,42,37,43)(35,40,46,56,47,55,53,60)(36,48,51,58,52,41,50,59)(45,61,49,64,62,54,63,57);(1,3,5,14,7,15,6,22)(2,8,10,12,11,25,21,24)(4,17,18,31,20,13,19,30)(9,26,16,29,27,23,28,32)(33,35)(34,40)(36,49)(37,53)(38,46)(39,47)(41,57)(42,55)(43,60)(44,56)(45,50)(48,64)(51,62)(52,63)(54,58)(59,61);(1,4)(2,9)(3,13)(5,19)(6,18)(7,20)(8,23)(10,28)(11,27)(12,29)(14,31)(15,17)(16,21)(22,30)(24,32)(25,26)(33,36,39,52)(34,41,42,48)(35,45,47,62)(37,51,38,50)(40,54,55,61)(43,59,44,58)(46,63,53,49)(56,64,60,57)

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=i,d^2=h,e^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,c^b=cf,e^b=eh,f^b=fhi,d^c=dh,e^c=eh,f^c=fi,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, 24506);