#6343712

order	512
nilpotency class	3
isoclinism	isoclinic class of #6343554
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×C₂²⋊C₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#62831	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2661 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,e^2=h,f^2=i,b^a=bf,d^a=dh,e^a=ei,f^a=fh,f^b=fi,g^b=ghi,d^c=di,e^c=eh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6250908);