#263995

order	512
nilpotency class	3
isoclinism	isoclinic class of #263983
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
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

791 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)(4,9)(5,8)(6,10)(7,13)(12,16)(14,15)(18,23)(20,33)(21,36)(24,38)(25,41)(26,27)(28,40)(29,45)(30,31)(34,46)(35,47)(44,48);(1,3,4,13)(2,7,9,11)(5,14,6,12)(8,16,10,15)(17,18,21,23,22,26,36,27)(19,28,30,40,32,38,31,24)(20,25,33,37,35,41,47,43)(29,44,45,42,34,48,46,39);(2,8)(3,12)(5,6)(9,10)(13,14)(15,16)(17,19)(18,24)(20,34)(21,30)(22,32)(23,28)(25,42)(26,40)(27,38)(29,35)(31,36)(33,46)(37,48)(39,41)(43,44)(45,47);(2,9)(7,11)(8,10)(15,16)(17,20,21,33,22,35,36,47)(18,25,23,37,26,41,27,43)(19,29,30,45,32,34,31,46)(24,39,28,44,40,42,38,48)

Polycyclic presentation:

2^9|b^2=e,d^2=ei,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=dh,f^c=fi,g^c=ghi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 171191);