#118569

order	512
nilpotency class	3
isoclinism	isoclinic class of #118564
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#121	Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

513 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=h,e^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fi,c^b=cf,f^b=fi,d^c=di,e^c=ei,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, 25765);