#234916

order	512
nilpotency class	3
isoclinism	isoclinic class of #234906
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1150 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)(3,13,16,25)(4,17,18,26)(5,11,6,10)(8,20,23,15)(9,21,24,19)(14,22,28,32)(27,29,31,30)(33,34)(35,39)(36,40)(37,41)(38,42)(43,51)(44,52)(45,53)(46,55)(47,56)(48,57)(49,58)(50,59)(54,61)(60,63)(62,64);(1,3)(2,8)(4,14)(5,20)(6,15)(7,16)(9,22)(10,25)(11,13)(12,23)(17,31)(18,28)(19,30)(21,29)(24,32)(26,27)(33,35,38,45)(34,39,42,53)(36,43,47,61)(37,48,50,44)(40,54,56,51)(41,57,59,52)(46,62,49,60)(55,63,58,64);(1,4)(2,9)(3,14)(5,19)(6,21)(7,18)(8,22)(10,17)(11,26)(12,24)(13,27)(15,29)(16,28)(20,30)(23,32)(25,31)(33,36)(34,40)(35,43)(37,49)(38,47)(39,51)(41,58)(42,56)(44,62)(45,61)(46,50)(48,60)(52,64)(53,54)(55,59)(57,63);(4,18)(9,24)(14,28)(17,26)(19,21)(22,32)(27,31)(29,30)(33,37,38,50)(34,41,42,59)(35,44,45,48)(36,46,47,49)(39,52,53,57)(40,55,56,58)(43,60,61,62)(51,63,54,64)

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=i,e^2=h,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,g^a=gi,c^b=cg,d^b=di,e^b=eh,d^c=dhi,e^c=ei,f^c=fh

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, 142112);