#332838

order	512
nilpotency class	3
isoclinism	isoclinic class of #332711
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

550 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=i,d^2=hi,e^2=h,f^2=i,g^2=h,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,g^b=gh,f^c=fi,g^c=gh,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, 240034);