#129461

order	512
nilpotency class	4
isoclinism	isoclinic class of #129412
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	C₂×#535	C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#518 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

921 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 921 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=hi,d^2=i,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ch,d^b=di,f^b=fgi,e^c=ei,f^c=fi,f^d=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, 36657);