#107233

order	512
nilpotency class	3
isoclinism	isoclinic class of #107214
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

692 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)(4,15,17,8)(5,18)(6,10)(7,13)(9,20)(12,25,28,32)(14,26)(16,23,19,24)(21,27,31,29)(22,30)(33,34,37,43,39,42,38,52)(35,40,46,55,45,54,50,56)(36,47,48,58,49,41,51,57)(44,59,61,64,60,53,62,63);(1,3)(2,7,10,22)(4,12)(5,13)(6,14)(8,21,15,31)(9,11,18,26)(16,27)(17,28)(19,29)(20,30)(23,32,24,25)(33,35,38,46,39,45,37,50)(34,40,43,56,42,54,52,55)(36,44,48,62,49,60,51,61)(41,53,58,63,47,59,57,64);(1,4)(2,8)(3,12)(5,19)(6,17)(7,21)(9,24)(10,15)(11,25)(13,29)(14,28)(16,20)(18,23)(22,31)(26,32)(27,30)(33,36)(34,41)(35,44)(37,51)(38,48)(39,49)(40,53)(42,47)(43,58)(45,60)(46,62)(50,61)(52,57)(54,59)(55,64)(56,63)

Polycyclic presentation:

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

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, 14429);