#240917

order	512
nilpotency class	3
isoclinism	isoclinic class of #240913
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	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

777 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:

(1,2,7,12)(3,13,15,25)(4,18,21,9)(5,11,6,10)(8,17,24,16)(14,29,30,26)(19,32,31,23)(20,27,22,28)(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)(2,8)(4,19)(5,17)(6,16)(7,15)(9,26)(10,25)(11,13)(12,24)(14,22)(18,29)(20,30)(21,31)(23,28)(27,32)(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);(1,4,5,20,7,21,6,22)(2,9,10,27,12,18,11,28)(3,14,16,31,15,30,17,19)(8,23,13,29,24,32,25,26)(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);(3,15)(8,24)(13,25)(14,30)(16,17)(19,31)(23,32)(26,29)(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,g^2=h,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,g^b=gh,d^c=di,f^c=fhi

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, 148113);