#120647

order	512
nilpotency class	3
isoclinism	isoclinic class of #10557
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#394
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1442
prime radical	C₂⁴	C₄×Q₈
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

386 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 118 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=d,d^2=hi,e^2=hi,f^2=hi,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=cf,d^b=di,f^b=fh,g^b=gi,e^c=ei,f^c=fh,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 27843);