#120648

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

392 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 126 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=d,d^2=hi,e^2=hi,f^2=hi,g^2=i,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, 27844);