#345163

order	512
nilpotency class	3
isoclinism	isoclinic class of #334823
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

549 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 549 conjugacy classes of subgroups — too many to draw. Only its 144 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) = 512

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=efh,b^2=ehi,c^2=ehi,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=defh,c^b=cg,d^b=de,g^b=gi,f^c=fh,g^c=gi,f^d=fh

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, 252359);