#247830

order	512
nilpotency class	3
isoclinism	isoclinic class of #247783
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

958 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)(3,13)(4,9)(5,18)(6,21)(7,12)(8,15)(10,19)(11,23)(14,24)(16,32)(17,25)(20,28)(22,27)(26,31)(29,30)(33,34,38,42)(35,39,44,53)(36,46,48,56)(37,49,51,41)(40,50,55,47)(43,57,58,64)(45,61,60,54)(52,62,63,59);(1,3,7,4)(2,8,12,25)(5,14,20,16)(6,17,23,15)(9,21,13,11)(10,24,27,32)(18,29,28,26)(19,31,22,30)(33,35,38,44)(34,39,42,53)(36,43,48,58)(37,45,51,60)(40,52,55,63)(41,54,49,61)(46,57,56,64)(47,59,50,62);(1,4,7,3)(2,9,12,13)(5,16,20,14)(6,15,23,17)(8,11,25,21)(10,26,27,29)(18,24,28,32)(19,30,22,31)(33,36,38,48)(34,40,42,55)(35,43,44,58)(37,50,51,47)(39,52,53,63)(41,56,49,46)(45,62,60,59)(54,64,61,57);(1,5,7,20)(2,10,12,27)(3,14,4,16)(6,22,23,19)(8,24,25,32)(9,26,13,29)(11,28,21,18)(15,31,17,30)(35,44)(39,53)(43,58)(45,60)(52,63)(54,61)(57,64)(59,62)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=hi,d^2=h,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gh,c^b=cg,d^b=di,e^b=eh,e^c=eh,f^c=fi,e^d=eh,f^d=fh

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, 155026);