#263918

order	512
nilpotency class	3
isoclinism	isoclinic class of #263899
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	#3106	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

789 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)(4,9)(5,8)(6,10)(7,13)(12,16)(14,15)(17,18,23,28)(19,25,31,40)(20,32,36,47)(21,37,22,27)(24,38,41,30)(26,35,44,34)(29,43,33,39)(42,48,46,45);(1,3,4,13)(2,7,9,11)(5,14,6,12)(8,16,10,15)(17,19,22,38,23,31,21,30)(18,24,37,40,28,41,27,25)(20,33,35,45,36,29,34,48)(26,43,47,42,44,39,32,46);(2,8)(3,12)(5,6)(9,10)(13,14)(15,16)(17,19)(18,25)(20,29)(21,30)(22,38)(23,31)(24,37)(26,42)(27,41)(28,40)(32,43)(33,36)(34,45)(35,48)(39,47)(44,46);(2,9)(7,11)(8,10)(15,16)(17,20,21,34,23,36,22,35)(18,26,27,32,28,44,37,47)(19,29,30,45,31,33,38,48)(24,39,25,42,41,43,40,46)

Polycyclic presentation:

2^9|a^2=h,b^2=eh,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,c^b=cg,d^b=dh,g^b=ghi,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, 171114);