#6355481

order	512
nilpotency class	3
isoclinism	isoclinic class of #6355448
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	#60967	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1506 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)(3,12)(4,9)(5,10)(6,20)(7,11)(8,16)(13,25)(14,26)(15,22)(17,23)(18,24)(19,30)(21,28)(27,31)(29,32)(33,34,39,42)(35,45,49,56)(36,50,51,58)(37,44,38,43)(40,48,55,47)(41,53,57,52)(46,60,61,54)(59,63,64,62);(1,3,6,16,7,14,5,15)(2,8,10,12,11,22,20,26)(4,13,19,28,17,29,18,27)(9,21,24,25,23,31,30,32)(33,35,37,47,39,49,38,48)(34,40,43,45,42,55,44,56)(36,46,52,62,51,61,53,63)(41,54,50,59,57,60,58,64);(1,4)(2,9)(3,13)(5,18)(6,19)(7,17)(8,21)(10,24)(11,23)(12,25)(14,29)(15,27)(16,28)(20,30)(22,31)(26,32)(33,36)(34,41)(35,46)(37,52)(38,53)(39,51)(40,54)(42,57)(43,50)(44,58)(45,59)(47,62)(48,63)(49,61)(55,60)(56,64);(3,14)(4,17)(8,22)(9,23)(12,26)(15,16)(18,19)(24,30)(36,51)(41,57)(46,61)(50,58)(52,53)(54,60)(59,64)(62,63);(1,5,7,6)(2,10,11,20)(3,15,14,16)(4,18,17,19)(8,12,22,26)(9,24,23,30)(13,27,29,28)(21,25,31,32)(34,42)(40,55)(41,57)(43,44)(45,56)(50,58)(54,60)(59,64)

Polycyclic presentation:

2^9|a^2=i,b^2=f,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fh,g^a=gi,d^b=dhi,d^c=dh,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306848, #72306850, #72306925
maximal normal subgroup of	#72306848, #72306850, #72306925

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6262677);