#348027

order	512
nilpotency class	3
isoclinism	isoclinic class of #348018
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	#537	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 777 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=eh,d^2=i,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,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, 255223);