#6354256

order	512
nilpotency class	3
isoclinism	isoclinic class of #6354237
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	#60962	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1621 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)(4,10)(5,7)(6,9)(8,12)(13,16)(14,15)(17,18)(19,28)(20,25)(21,26)(22,36)(23,27)(24,31)(29,43)(30,39)(32,38)(33,40)(34,41)(35,42)(37,45)(44,47)(46,48);(2,7)(8,15)(9,10)(11,16)(17,19,22,31,23,32,21,30)(18,24,26,28,27,39,36,38)(20,29,34,45,35,46,33,44)(25,37,40,43,42,47,41,48);(1,3)(2,8)(4,12)(5,13)(6,14)(7,15)(9,11)(10,16)(17,20)(18,25)(19,29)(21,33)(22,34)(23,35)(24,37)(26,40)(27,42)(28,43)(30,44)(31,45)(32,46)(36,41)(38,48)(39,47);(1,4,5,6)(2,9,7,10)(3,12,13,14)(8,11,15,16)(17,21,23,22)(18,26,27,36)(19,30,32,31)(20,33,35,34)(24,28,39,38)(25,40,42,41)(29,44,46,45)(37,43,47,48);(17,21,23,22)(18,26,27,36)(19,31,32,30)(20,34,35,33)(24,38,39,28)(25,41,42,40)(29,44,46,45)(37,43,47,48)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306705, #72306708, #72306742
maximal normal subgroup of	#72306705, #72306708, #72306742

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6261452);