#6354768

order	512
nilpotency class	3
isoclinism	isoclinic class of #6354705
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	C₂²×#41	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1604 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:

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

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,f^a=fh,g^a=gi,d^b=di,e^b=eh,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306795, #72306852
maximal normal subgroup of	#72306795, #72306852

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6261964);