#108015

order	512
nilpotency class	3
isoclinism	isoclinic class of #108006
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	C₂×#121	Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

689 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)(3,13,15,8,16,23,17,24)(4,18,19,25,20,9,21,26)(14,27,28,31,29,32,30,22)(33,34,38,42,39,43,37,53)(35,44,46,40,48,55,47,56)(36,49,51,41,52,57,50,58)(45,60,61,59,62,54,63,64);(1,3,6,17,7,16,5,15)(2,8,12,23,11,24,10,13)(4,14,19,28,20,29,21,30)(9,22,25,32,18,31,26,27)(33,35)(34,40)(36,45)(37,47)(38,46)(39,48)(41,54)(42,55)(43,56)(44,53)(49,59)(50,63)(51,61)(52,62)(57,64)(58,60);(1,4)(2,9)(3,14)(5,21)(6,19)(7,20)(8,22)(10,26)(11,18)(12,25)(13,27)(15,30)(16,29)(17,28)(23,32)(24,31)(33,36)(34,41)(35,45)(37,50)(38,51)(39,52)(40,54)(42,57)(43,58)(44,59)(46,61)(47,63)(48,62)(49,53)(55,64)(56,60)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 15211);