#118198

order	512
nilpotency class	3
isoclinism	isoclinic class of #118195
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	C₂×#121	Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

522 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)(33,34,38,44,39,42,37,43)(35,40,48,56,46,57,47,55)(36,49,53,59,51,41,52,60)(45,54,62,63,61,64,50,58);(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)(36,50,51,62)(37,47)(38,48)(39,46)(41,58,49,63)(42,57)(43,55)(44,56)(45,52,61,53)(54,59,64,60);(1,4)(2,9)(3,14)(5,19)(6,22)(7,20)(8,23)(10,26)(11,17)(12,28)(13,29)(15,30)(16,31)(18,21)(24,32)(25,27)(33,36)(34,41)(35,45)(37,53)(38,52)(39,51)(40,54)(42,49)(43,60)(44,59)(46,61)(47,62)(48,50)(55,63)(56,58)(57,64)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 25394);