#130460

order	512
nilpotency class	4
isoclinism	isoclinic class of #130252
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1786
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#418
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

511 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 511 conjugacy classes of subgroups — too many to draw. Only its 96 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) = 256

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=g,d^2=hi,e^2=i,f^2=ghi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,d^b=dh,e^b=eg,f^b=fgh,h^b=hi,d^c=dg,e^c=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 37656);