#236842

order	512
nilpotency class	3
isoclinism	isoclinic class of #236825
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	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

1078 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)(3,13,16,24)(4,9,19,26)(5,11,6,10)(8,17,23,15)(14,29,30,25)(18,32,31,22)(20,28,21,27)(33,34)(35,40)(36,48)(37,42)(38,51)(39,43)(41,49)(44,59)(45,54)(46,58)(47,55)(50,56)(52,57)(53,60)(61,63)(62,64);(1,3)(2,8)(4,18)(5,17)(6,15)(7,16)(9,25)(10,24)(11,13)(12,23)(14,20)(19,31)(21,30)(22,27)(26,29)(28,32)(33,35,39,47)(34,40,43,55)(36,44,50,61)(37,46,38,45)(41,53,57,64)(42,58,51,54)(48,63,56,59)(49,62,52,60);(1,4,7,19)(2,9,12,26)(3,14,16,30)(5,20,6,21)(8,22,23,32)(10,27,11,28)(13,25,24,29)(15,18,17,31)(33,36)(34,41)(35,44)(37,49)(38,52)(39,50)(40,53)(42,56)(43,57)(45,60)(46,62)(47,61)(48,51)(54,63)(55,64)(58,59);(4,19)(9,26)(14,30)(18,31)(20,21)(22,32)(25,29)(27,28)(33,37,39,38)(34,42,43,51)(35,45,47,46)(36,49,50,52)(40,54,55,58)(41,56,57,48)(44,60,61,62)(53,63,64,59)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,d^2=i,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=gh,c^b=cg,d^b=di,e^b=eh,f^b=fhi,g^b=gh,d^c=dh,e^c=ei,f^c=fi,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, 144038);