#6354137

order	512
nilpotency class	3
isoclinism	isoclinic class of #6354110
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	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1550 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,11)(4,9)(5,8)(6,10)(7,12)(13,15)(14,16)(17,18,22,26)(19,27,30,41)(20,24,32,38)(21,34,28,25)(23,35,37,31)(29,40,43,48)(33,45,42,39)(36,46,47,44);(17,19,28,35,22,30,21,31)(18,23,34,41,26,37,25,27)(20,29,42,46,32,43,33,44)(24,36,45,48,38,47,39,40);(1,3)(2,7)(4,12)(5,14)(6,13)(8,11)(9,15)(10,16)(17,20)(18,24)(19,29)(21,33)(22,32)(23,36)(25,39)(26,38)(27,40)(28,42)(30,43)(31,44)(34,45)(35,46)(37,47)(41,48);(1,4,6,5)(2,8,10,9)(3,12,13,14)(7,11,16,15)(20,32)(24,38)(29,43)(33,42)(36,47)(39,45)(40,48)(44,46);(1,4,6,5)(2,9,10,8)(3,12,13,14)(7,15,16,11)(19,30)(23,37)(27,41)(29,43)(31,35)(36,47)(40,48)(44,46)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306670, #72306673, #72306719
maximal normal subgroup of	#72306670, #72306673, #72306719

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6261333);