#348096

order	512
nilpotency class	3
isoclinism	isoclinic class of #348090
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3529	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

775 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,g^a=gh,c^b=ch,f^c=fi,g^c=ghi,f^d=fhi,g^d=gh

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, 255292);