#348648

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

Polycyclic presentation:

2^9|a^2=i,b^2=eh,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, 255844);