#112198

order	512
nilpotency class	3
isoclinism	isoclinic class of #112191
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	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

730 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=i,e^2=h,f^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,c^b=cf,d^b=di,f^b=fh,g^b=ghi,d^c=dhi,e^c=eh,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, 19394);