#245563

order	512
nilpotency class	3
isoclinism	isoclinic class of #245519
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1047 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 152759);