#214634

order	512
nilpotency class	3
isoclinism	isoclinic class of #214020
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 864 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=hi,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,d^c=di,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, 121830);