#112386

order	512
nilpotency class	3
isoclinism	isoclinic class of #112377
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	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

733 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)(4,8)(5,10)(6,9)(12,29,30,21)(13,27,18,22)(15,32,31,23)(16,25)(17,24)(19,28)(20,26)(33,34,38,44,39,43,37,42)(35,45,48,40,49,56,47,55)(36,50,51,58,52,41,53,57)(46,60,61,54,62,59,63,64);(1,3)(2,7)(4,15)(5,18)(6,14)(8,23)(9,11)(10,27)(12,16)(13,20)(17,31)(19,30)(21,25)(22,26)(24,32)(28,29)(33,35,39,49)(34,40,43,55)(36,46,52,62)(37,48,38,47)(41,54,50,64)(42,56,44,45)(51,63,53,61)(57,60,58,59);(1,4,6,17)(2,8,9,24)(3,12,14,30)(5,19,20,16)(7,21,11,29)(10,28,26,25)(13,31,18,15)(22,32,27,23)(33,36)(34,41)(35,46)(37,51)(38,53)(39,52)(40,54)(42,57)(43,50)(44,58)(45,59)(47,61)(48,63)(49,62)(55,64)(56,60)

Polycyclic presentation:

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

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, 19582);