#108028

order	512
nilpotency class	3
isoclinism	isoclinic class of #108026
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	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

703 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)(3,11,13,7)(4,15,16,8)(5,10)(6,9)(12,23)(14,24,27,20)(17,28,29,21)(18,22)(19,25)(26,31)(30,32)(33,34)(35,43)(36,47,48,57)(37,41)(38,42)(39,45)(40,49,55,51)(44,60,62,64)(46,56)(50,54)(52,58)(53,63,59,61);(1,3,5,14,6,13,18,27)(2,7,10,20,9,11,22,24)(4,12,17,26,16,25,29,32)(8,19,21,30,15,23,28,31)(33,35)(34,39,42,43)(36,44)(37,50)(38,45)(40,53,55,59)(41,56,58,54)(46,52)(47,60,57,64)(48,62)(49,63)(51,61);(1,4)(2,8)(3,12)(5,17)(6,16)(7,19)(9,15)(10,21)(11,23)(13,25)(14,26)(18,29)(20,30)(22,28)(24,31)(27,32)(33,36)(34,40)(35,44)(37,51)(38,48)(39,53)(41,57)(42,55)(43,59)(45,62)(46,63)(47,58)(49,52)(50,61)(54,60)(56,64)

Polycyclic presentation:

2^9|a^2=f,b^2=g,e^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gi,d^b=di,e^b=ehi,f^b=fh,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, 15224);