#332710

order	512
nilpotency class	3
isoclinism	isoclinic class of #332583
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

557 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:

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

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 239906);