#108301

order	512
nilpotency class	3
isoclinism	isoclinic class of #108282
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²⋊#95
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

681 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,d^2=h,e^2=i,f^2=hi,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gh,e^b=ei,d^c=di,e^c=ei,f^c=fhi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 15497);