#333005

order	512
nilpotency class	3
isoclinism	isoclinic class of #332983
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2669
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3506	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

660 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)(4,7)(5,9)(6,11)(8,14)(12,16)(13,15)(17,18,21,27,23,29,22,28)(19,30,32,42,34,41,33,24)(20,25,36,26,37,44,38,45)(31,43,35,46,47,40,48,39);(17,19,23,34)(18,24,29,42)(20,35,37,48)(21,33,22,32)(25,43,44,40)(26,39,45,46)(27,41,28,30)(31,38,47,36);(1,3,5,12)(2,6,9,15)(4,13,14,11)(7,10,8,16)(17,20)(18,25)(19,31)(21,36)(22,38)(23,37)(24,39)(26,27)(28,45)(29,44)(30,43)(32,35)(33,48)(34,47)(40,41)(42,46);(2,7)(4,14)(6,10)(8,9)(11,13)(15,16)(17,20)(18,26)(19,31)(21,38)(22,36)(23,37)(24,40)(25,27)(28,44)(29,45)(30,46)(32,48)(33,35)(34,47)(39,41)(42,43)

Polycyclic presentation:

2^9|a^2=e,b^2=h,c^2=i,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=defh,f^a=fi,c^b=cg,d^b=de,e^b=eh,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, 240201);