#112236

order	512
nilpotency class	3
isoclinism	isoclinic class of #112221
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

750 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)(3,12,16,8)(4,17,21,28)(5,11)(6,22)(7,10)(9,19,26,20)(13,30)(14,27,15,24)(18,29)(23,31)(25,32)(33,34,38,44,39,43,37,42)(35,45,48,40,49,55,47,54)(36,41,50,57,51,58,52,59)(46,60,61,53,62,64,63,56);(1,3)(2,8)(4,18)(5,15)(6,14)(7,16)(9,25)(10,12)(11,27)(13,20)(17,23)(19,32)(21,31)(22,24)(26,30)(28,29)(33,35,39,49)(34,40,43,54)(36,46,51,62)(37,48,38,47)(41,56,58,53)(42,55,44,45)(50,63,52,61)(57,64,59,60);(1,4,7,21)(2,9,10,26)(3,13,16,32)(5,20,6,19)(8,23,12,29)(11,28,22,17)(14,18,15,31)(24,30,27,25)(33,36)(34,41)(35,46)(37,52)(38,50)(39,51)(40,53)(42,59)(43,58)(44,57)(45,60)(47,63)(48,61)(49,62)(54,56)(55,64)

Polycyclic presentation:

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

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, 19432);