#142295

order	512
nilpotency class	4
isoclinism	isoclinic class of #142240
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1999
commutator, Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₄⋊C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

608 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 608 conjugacy classes of subgroups — too many to draw. Only its 72 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)(3,5)(4,6)(7,8)(9,10,16,22)(11,23,28,39)(12,19,30,35)(13,18,29,36)(14,21,15,20)(17,31,34,25)(24,38,40,33)(26,32,27,37);(5,8)(9,11)(10,17)(12,25)(13,24)(14,26)(15,27)(16,28)(18,23)(19,37)(20,33)(21,38)(22,34)(29,40)(30,31)(32,35)(36,39);(1,3)(2,5)(4,7)(6,8)(9,12,15,29,16,30,14,13)(10,18,21,35,22,36,20,19)(11,24,26,31,28,40,27,25)(17,32,33,23,34,37,38,39)

Polycyclic presentation:

2^9|a^2=i,c^2=g,d^2=gi,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fh,g^a=gi,c^b=cf,d^b=dg,f^b=fi,g^b=gi,d^c=dh

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, 49491);