#264041

order	512
nilpotency class	3
isoclinism	isoclinic class of #264023
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#62558	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

781 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)(3,11)(4,12)(5,8)(6,10)(7,14)(9,15)(13,16)(18,23)(20,33)(21,35)(24,38)(25,41)(26,27)(28,40)(29,45)(30,31)(34,46)(36,47)(44,48);(1,3,5,12)(2,4,8,11)(6,7,15,13)(9,14,10,16)(17,18,21,23,22,26,35,27)(19,28,30,40,32,38,31,24)(20,25,33,37,36,41,47,43)(29,44,45,42,34,48,46,39);(1,4,5,11)(2,7,8,13)(3,10,12,9)(6,16,15,14)(17,19)(18,24)(20,34)(21,30)(22,32)(23,28)(25,42)(26,40)(27,38)(29,36)(31,35)(33,46)(37,48)(39,41)(43,44)(45,47);(2,8)(4,11)(9,10)(14,16)(17,20,35,47,22,36,21,33)(18,25,27,43,26,41,23,37)(19,29,31,46,32,34,30,45)(24,39,38,48,40,42,28,44)

Polycyclic presentation:

2^9|b^2=e,c^2=i,d^2=ehi,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gh,c^b=cg,f^b=fh,g^b=ghi,d^c=dhi,f^c=fi,g^c=ghi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 171237);