#124991

order	512
nilpotency class	4
isoclinism	isoclinic class of #124826
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#99
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂×#95	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂×#95 ▷ C₂⁴ ▷ C₂ ▷ C₁

510 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 510 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=gi,d^2=g,e^2=h,b^a=bd,c^a=cfgh,d^a=df,f^a=fi,d^b=dg,e^b=efg,e^c=ei,e^d=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, 32187);