#332474

order	512
nilpotency class	3
isoclinism	isoclinic class of #332455
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3100	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

603 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|b^2=h,d^2=hi,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=def,e^a=eh,f^a=fi,g^a=gh,c^b=cg,d^b=dei,e^b=eh,f^b=fh,f^c=fi,e^d=eh,f^d=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 239670);