#244519

order	512
nilpotency class	3
isoclinism	isoclinic class of #244499
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#61765	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

994 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=h,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,g^a=gi,c^b=cg,e^b=eh,g^b=gh,e^c=ei,g^c=gh,e^d=eh,f^d=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151715);