#142351

order	512
nilpotency class	4
isoclinism	isoclinic class of #142236
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#447
commutator, Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#408 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 464 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) = 52
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 52 points:

(1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(13,16)(14,17)(15,19)(18,20)(21,22,27,33)(23,29,36,47)(24,28,40,44)(25,32,26,31)(30,39,43,37)(34,49,41,46)(35,51,38,42)(45,52,48,50);(5,7,9,14)(6,10,12,17)(8,13,15,20)(11,18,19,16)(22,28)(23,34)(24,37)(25,26)(29,45)(30,32)(31,43)(33,44)(35,38)(36,41)(39,40)(42,46)(47,48)(49,51);(2,4)(5,8)(6,11)(7,13)(9,15)(10,16)(12,19)(14,20)(17,18)(21,23)(22,29)(24,38)(25,41)(26,34)(27,36)(28,42)(30,48)(31,49)(32,46)(33,47)(35,40)(37,52)(39,50)(43,45)(44,51)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,d^2=gi,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,f^a=fh,g^a=gi,c^b=cf,d^b=dg,f^b=fi,g^b=gi,d^c=dh,f^c=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, 49547);