#348046

order	512
nilpotency class	3
isoclinism	isoclinic class of #348038
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3459	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

828 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 828 conjugacy classes of subgroups — too many to draw. Only its 156 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 255242);