#214611

order	512
nilpotency class	3
isoclinism	isoclinic class of #214000
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

956 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 956 conjugacy classes of subgroups — too many to draw. Only its 190 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) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=i,f^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=ce,f^b=fh,g^b=gh,f^c=fh,g^c=ghi,g^d=gh

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, 121807);