#263968

order	512
nilpotency class	3
isoclinism	isoclinic class of #263963
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	#3171	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

740 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=e,d^2=eh,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,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, 171164);