#111980

order	512
nilpotency class	3
isoclinism	isoclinic class of #111967
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	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

767 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fhi,c^b=cf,d^b=di,e^b=eh,g^b=ghi,d^c=di,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, 19176);