#115361

order	512
nilpotency class	3
isoclinism	isoclinic class of #115356
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	C₂×#121	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

752 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)(3,13)(4,18,22,26)(5,11)(6,10)(7,12)(8,15)(9,21,27,20)(14,23,30,32)(16,24)(17,28)(19,25,31,29)(33,34,37,41,38,42,48,54)(35,39,45,51,44,50,58,62)(36,40,47,53,46,52,59,63)(43,55,57,64,56,49,60,61);(1,3)(2,8)(4,19)(5,16)(6,15)(7,17)(9,25)(10,28)(11,13)(12,24)(14,20)(18,32)(21,30)(22,31)(23,26)(27,29)(33,35)(34,39)(36,43)(37,45)(38,44)(40,49)(41,51)(42,50)(46,56)(47,57)(48,58)(52,55)(53,61)(54,62)(59,60)(63,64);(1,4,7,22)(2,9,12,27)(3,14,17,30)(5,20,6,21)(8,23,24,32)(10,18,11,26)(13,29,28,25)(15,19,16,31)(33,36)(34,40)(35,43)(37,47)(38,46)(39,49)(41,53)(42,52)(44,56)(45,57)(48,59)(50,55)(51,61)(54,63)(58,60)(62,64)

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=h,e^2=h,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,c^b=cf,d^b=dh,e^b=ei,f^b=fh,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, 22557);