#236699

order	512
nilpotency class	3
isoclinism	isoclinic class of #236695
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	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1095 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)(4,17)(5,22)(6,10)(7,11)(8,14)(9,21)(13,30)(15,25)(16,24)(18,29)(19,28)(20,27)(23,31)(26,32)(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)(2,8)(4,18)(5,15)(6,14)(7,16)(9,26)(10,24)(11,25)(12,22)(13,20)(17,23)(19,32)(21,31)(27,29)(28,30)(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);(1,4,7,19)(2,9,11,27)(3,13,16,31)(5,20,6,21)(8,23,25,30)(10,28,22,17)(12,29,24,26)(14,18,15,32)(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);(4,19)(9,27)(13,31)(17,28)(18,32)(20,21)(23,30)(26,29)(35,44)(36,47)(39,54)(40,56)(45,50)(46,58)(48,51)(55,57)

Polycyclic presentation:

2^9|a^2=i,c^2=h,e^2=h,f^2=hi,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fhi,c^b=cg,d^b=di,e^b=eh,f^b=fi,g^b=gh,d^c=dhi,e^c=ei,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, 143895);