#100838

order	512
nilpotency class	3
isoclinism	isoclinic class of #100827
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

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

normal subgroups only

This group has 378 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfi,c^2=gi,d^2=gh,e^2=ghi,b^a=bd,c^a=ce,d^a=dg,e^a=egh,d^b=dh,e^b=ei,f^b=fh,d^c=di,e^c=eghi,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, 8034);