#130469

order	512
nilpotency class	4
isoclinism	isoclinic class of #130242
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	#466	C₂×C₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#446 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

526 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 92 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)(17,18,22,28,20,29,36,26,23,27,39,44,37,25,21,38)(19,30,34,42,35,43,48,46,33,45,47,24,31,41,32,40);(1,2)(3,7)(4,12)(5,8)(6,11)(9,14)(10,16)(13,15)(17,19,23,33)(18,24,27,42)(20,31,37,35)(21,34,36,47)(22,32,39,48)(25,40,29,46)(26,43,38,41)(28,45,44,30);(1,3,5,10)(2,7,8,16)(4,13,11,9)(6,14,12,15)(17,20,23,37)(18,25,27,29)(19,31,33,35)(21,39,36,22)(24,40,42,46)(26,44,38,28)(30,41,45,43)(32,34,48,47)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=g,d^2=hi,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,f^c=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 37665);