#236588

order	512
nilpotency class	3
isoclinism	isoclinic class of #236579
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

1093 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)(3,12)(4,9)(5,20)(6,10)(7,11)(8,14)(13,29)(15,23)(16,28)(17,22)(18,25)(19,27)(21,26)(24,30)(31,32)(33,34,38,42)(35,39,44,54)(36,46,47,58)(37,49,52,41)(40,51,56,48)(43,59,60,63)(45,57,50,55)(53,62,64,61);(1,3,7,16)(2,8,11,23)(4,17,18,31)(5,15,6,14)(9,24,25,29)(10,28,20,12)(13,21,30,19)(22,26,32,27)(33,35)(34,39)(36,43)(37,50)(38,44)(40,53)(41,57)(42,54)(45,52)(46,63)(47,60)(48,62)(49,55)(51,61)(56,64)(58,59);(1,4,7,18)(2,9,11,25)(3,13,16,30)(5,19,6,21)(8,22,23,32)(10,26,20,27)(12,29,28,24)(14,17,15,31)(33,36)(34,40)(35,43)(37,51)(38,47)(39,53)(41,58)(42,56)(44,60)(45,62)(46,49)(48,52)(50,61)(54,64)(55,63)(57,59);(4,18)(9,25)(13,30)(17,31)(19,21)(22,32)(24,29)(26,27)(35,44)(36,47)(39,54)(40,56)(45,50)(46,58)(48,51)(55,57)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=cg,d^b=di,e^b=eh,f^b=fi,g^b=gh,d^c=dhi,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, 143784);