#233875

order	512
nilpotency class	3
isoclinism	isoclinic class of #233856
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	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1115 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)(3,13,17,24)(4,9,21,27)(5,10,6,11)(8,15,23,16)(14,29,30,25)(18,32,31,22)(19,26,20,28)(33,34,39,44)(35,45,49,55)(36,50,51,58)(37,43,38,42)(40,48,56,47)(41,52,57,53)(46,60,61,54)(59,62,64,63);(1,3,7,17)(2,8,12,23)(4,18,21,31)(5,15,6,16)(9,25,27,29)(10,13,11,24)(14,19,30,20)(22,28,32,26)(33,35,39,49)(34,40,44,56)(36,46,51,61)(37,48,38,47)(41,54,57,60)(42,55,43,45)(50,64,58,59)(52,63,53,62);(1,4)(2,9)(3,14)(5,20)(6,19)(7,21)(8,22)(10,28)(11,26)(12,27)(13,25)(15,31)(16,18)(17,30)(23,32)(24,29)(33,36)(34,41)(35,46)(37,53)(38,52)(39,51)(40,54)(42,50)(43,58)(44,57)(45,59)(47,63)(48,62)(49,61)(55,64)(56,60);(1,5,7,6)(2,10,12,11)(3,15,17,16)(4,19,21,20)(8,13,23,24)(9,26,27,28)(14,18,30,31)(22,29,32,25)(36,51)(41,57)(46,61)(50,58)(52,53)(54,60)(59,64)(62,63)

Polycyclic presentation:

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

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, 141071);