#112136

order	512
nilpotency class	3
isoclinism	isoclinic class of #112133
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

745 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)(4,15,17,8)(5,19)(6,9)(10,22)(12,23)(13,24,20,27)(16,32)(18,26,21,28)(25,31)(29,30)(33,34)(35,43,46,57)(36,40)(37,41)(38,42)(39,45,53,49)(44,60,62,64)(47,55)(48,56)(50,58)(51,59)(52,63,54,61);(1,3,6,14)(2,7,9,11)(4,16,17,31)(5,20,22,13)(8,25,15,32)(10,27,19,24)(12,21,30,18)(23,28,29,26)(33,35)(34,39)(36,44)(37,49)(38,46)(40,54)(41,57)(42,53)(43,59)(45,51)(47,63)(48,62)(50,61)(52,56)(55,64)(58,60);(1,4)(2,8)(3,12)(5,21)(6,17)(7,23)(9,15)(10,28)(11,29)(13,31)(14,30)(16,20)(18,22)(19,26)(24,32)(25,27)(33,36,38,48)(34,40,42,56)(35,44,46,62)(37,50,51,47)(39,52,53,54)(41,58,59,55)(43,60,57,64)(45,63,49,61)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,d^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fhi,c^b=cf,d^b=di,f^b=fh,g^b=ghi,d^c=dhi,f^c=fh,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, 19332);