#333008

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	#3036	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

657 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)(17,18,23,24,22,28,21,27)(19,29,37,40,32,25,30,38)(20,33,36,26,35,42,34,41)(31,45,44,43,47,48,46,39);(18,24)(19,30)(20,34)(21,23)(26,41)(27,28)(32,37)(35,36)(38,40)(39,48)(43,45)(44,46);(1,3,5,12)(2,6,9,15)(4,13,14,11)(7,10,8,16)(17,19,22,32)(18,25,28,29)(20,31,35,47)(21,30,23,37)(24,38,27,40)(26,39,41,43)(33,48,42,45)(34,46,36,44);(2,7)(4,14)(6,10)(8,9)(11,13)(15,16)(17,20)(18,26)(19,31)(21,36)(22,35)(23,34)(24,33)(25,39)(27,42)(28,41)(29,43)(30,44)(32,47)(37,46)(38,48)(40,45)

Polycyclic presentation:

2^9|a^2=eh,c^2=hi,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, 240204);