#101654

order	512
nilpotency class	3
isoclinism	isoclinic class of #101519
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

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

normal subgroups only

This group has 392 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) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=gh,d^2=gh,e^2=gh,f^2=hi,b^a=bd,c^a=ce,d^a=dg,e^a=eghi,d^b=dh,e^b=ei,f^b=fh,d^c=di,e^c=egh,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, 8850);