#115532

order	512
nilpotency class	3
isoclinism	isoclinic class of #115519
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

740 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=h,f^2=hi,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,c^b=cf,d^b=dh,f^b=fhi,d^c=di,f^c=fhi,g^c=gh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 22728);