#107166

order	512
nilpotency class	3
isoclinism	isoclinic class of #107154
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

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	Q₈
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)(3,13)(4,18,20,9)(5,12)(6,10)(7,11)(8,15)(14,28,30,27)(16,23)(17,24)(19,25,21,26)(22,29,32,31)(33,34,37,41,38,42,49,55)(35,43,46,57,45,39,50,52)(36,40,48,54,47,53,60,63)(44,51,59,62,58,56,61,64);(1,3,6,16,7,17,5,15)(2,8,12,24,11,23,10,13)(4,14,21,31,20,30,19,29)(9,22,26,28,18,32,25,27)(33,35)(34,39,42,43)(36,44)(37,46)(38,45)(40,51,53,56)(41,52,55,57)(47,58)(48,59)(49,50)(54,62,63,64)(60,61);(1,4)(2,9)(3,14)(5,19)(6,21)(7,20)(8,22)(10,25)(11,18)(12,26)(13,27)(15,29)(16,31)(17,30)(23,32)(24,28)(33,36)(34,40)(35,44)(37,48)(38,47)(39,51)(41,54)(42,53)(43,56)(45,58)(46,59)(49,60)(50,61)(52,62)(55,63)(57,64)

Polycyclic presentation:

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

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, 14362);