#142353

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1999
commutator, Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁵	D₈
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 510 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) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,d^2=gi,g^2=i,b^a=bd,c^a=ce,d^a=dg,f^a=fhi,g^a=gi,c^b=cf,d^b=dg,e^b=ei,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, 49549);