#265096

order	512
nilpotency class	3
isoclinism	isoclinic class of #265019
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	C₂×#547	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

865 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)(4,7)(5,10)(6,9)(8,14)(12,15)(13,16)(17,18,24,30)(19,31,34,27)(20,35,38,46)(21,25,33,42)(22,29,23,28)(26,40,44,37)(32,47,48,41)(36,45,39,43);(2,7)(3,12)(5,6)(8,11)(13,14)(15,16)(17,19,23,33,24,34,22,21)(18,25,29,27,30,42,28,31)(20,32,40,36,38,48,37,39)(26,41,46,45,44,47,35,43);(1,3,4,13)(2,8,7,15)(5,12,6,14)(9,16,10,11)(17,20,24,38)(18,26,30,44)(19,32,34,48)(21,39,33,36)(22,37,23,40)(25,41,42,47)(27,45,31,43)(28,35,29,46);(2,7)(3,13)(9,10)(12,14)(17,21,22,34,24,33,23,19)(18,27,28,25,30,31,29,42)(20,36,37,32,38,39,40,48)(26,43,35,47,44,45,46,41)

Polycyclic presentation:

2^9|a^2=h,b^2=ei,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=ghi,c^b=cg,f^b=fhi,g^b=ghi,d^c=di,f^c=fi,g^c=ghi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 172292);