#118035

order	512
nilpotency class	3
isoclinism	isoclinic class of #113366
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#390
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1439
prime radical	C₂×#121	Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

526 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:

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

Polycyclic presentation:

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

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, 25231);