#333193

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	C₂³	#560
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

565 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=hi,c^2=i,d^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, 240389);