#333133

order	512
nilpotency class	3
isoclinism	isoclinic class of #333127
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	#3548	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

609 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=h,c^2=h,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,e^b=eh,g^b=ghi,f^c=fi,g^c=ghi,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, 240329);