#113972

order	512
nilpotency class	3
isoclinism	isoclinic class of #10128
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#392
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1438
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

751 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=i,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fi,c^b=cf,d^b=di,f^b=fh,g^b=ghi,d^c=di,e^c=ei,f^c=fh,g^c=gh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 21168);