#219494

order	512
nilpotency class	3
isoclinism	isoclinic class of #219491
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2328
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₈	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

592 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 592 conjugacy classes of subgroups — too many to draw. Only its 170 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=efh,d^2=ei,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=ce,f^b=fh,d^c=di,g^c=gh

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, 126690);