#130472

order	512
nilpotency class	4
isoclinism	isoclinic class of #130246
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	#62384	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 539 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)(17,18,22,25,24,30,20,36,23,29,37,43,21,28,39,27)(19,31,34,47,33,46,32,42,35,48,41,44,40,26,38,45);(1,2)(3,7)(4,12)(5,8)(6,11)(9,14)(10,16)(13,15)(18,25)(20,37)(21,24)(22,39)(26,42)(27,30)(28,36)(29,43)(31,47)(32,41)(33,40)(34,38)(44,48)(45,46);(1,3,5,10)(2,7,8,16)(4,13,11,9)(6,14,12,15)(17,19,23,35)(18,26,29,46)(20,38,39,32)(21,40,24,33)(22,41,37,34)(25,42,43,45)(27,47,36,44)(28,48,30,31)

Polycyclic presentation:

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

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, 37668);