#6368941

order	512
nilpotency class	3
isoclinism	isoclinic class of #6368846
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	#18726	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1452 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306703, #72306707, #72306720
maximal normal subgroup of	#72306703, #72306707, #72306720

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6276137);