#101668

order	512
nilpotency class	3
isoclinism	isoclinic class of #101519
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

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

373 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 373 conjugacy classes of subgroups — too many to draw. Only its 74 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=g,d^2=gh,e^2=gh,f^2=ghi,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, 8864);