#240725

order	512
nilpotency class	3
isoclinism	isoclinic class of #240721
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#394
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

764 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=e,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,d^b=dh,e^b=eh,f^b=fhi,d^c=di,f^c=fhi,g^c=gh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 147921);