#173648

order	512
nilpotency class	3
isoclinism	isoclinic class of #173481
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

625 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 625 conjugacy classes of subgroups — too many to draw. Only its 136 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) = 512

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=g,d^2=gh,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fi,d^b=dgh,e^b=eg,f^b=fh,e^c=eh,f^c=fi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 80844);