#233605

order	512
nilpotency class	3
isoclinism	isoclinic class of #232783
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#392
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1153 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 140801);