#118089

order	512
nilpotency class	3
isoclinism	isoclinic class of #118085
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	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

511 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)(3,13)(4,17,20,9)(5,12)(6,10)(7,11)(8,15)(14,29,31,25)(16,27)(18,32,30,23)(19,26,22,28)(21,24)(33,34,38,43,39,44,37,42)(35,40,48,56,46,54,47,55)(36,41,51,59,50,58,52,60)(45,61,49,64,62,53,63,57);(1,3,5,15,7,16,6,21)(2,8,10,13,11,24,12,27)(4,18,19,14,20,30,22,31)(9,25,26,23,17,29,28,32)(33,35)(34,40)(36,49,50,63)(37,47)(38,48)(39,46)(41,57,58,64)(42,55)(43,56)(44,54)(45,51,62,52)(53,59,61,60);(1,4,7,20)(2,9,11,17)(3,14,16,31)(5,19,6,22)(8,23,24,32)(10,26,12,28)(13,29,27,25)(15,30,21,18)(33,36)(34,41)(35,45)(37,52)(38,51)(39,50)(40,53)(42,60)(43,59)(44,58)(46,62)(47,63)(48,49)(54,61)(55,64)(56,57)

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=h,d^2=h,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fh,c^b=cf,e^b=ehi,f^b=fh,d^c=di,f^c=fhi,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, 25285);