#407023

order	512
nilpotency class	3
isoclinism	isoclinic class of #393900
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1185 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 1185 conjugacy classes of subgroups — too many to draw. Only its 218 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18)(19,28)(20,33)(21,38)(22,26)(23,27)(24,31)(25,36)(29,39)(30,41)(32,40)(34,42)(35,44)(37,43)(45,46)(47,48);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(17,19)(18,24)(20,34)(21,30)(22,31)(23,32)(25,42)(26,40)(27,41)(28,38)(29,35)(33,39)(36,47)(37,46)(43,48)(44,45);(1,4,5,3)(2,6,7,8)(17,20,23,37)(18,25,27,44)(19,29,32,47)(21,35,22,36)(24,39,41,48)(26,43,38,33)(28,45,40,42)(30,46,31,34);(10,14)(13,16)(17,21,23,22)(18,26,27,38)(19,30,32,31)(20,35,37,36)(24,40,41,28)(25,43,44,33)(29,46,47,34)(39,42,48,45)

Polycyclic presentation:

2^9|b^2=efh,c^2=ef,d^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 314219);