#108140

order	512
nilpotency class	3
isoclinism	isoclinic class of #108122
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	#3444	C₂²
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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,e^2=i,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,g^a=gi,d^b=di,e^b=eh,f^b=fh,d^c=dh,e^c=ei,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, 15336);