#120623

order	512
nilpotency class	3
isoclinism	isoclinic class of #10564
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×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 426 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=d,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=cf,d^b=dh,f^b=fi,g^b=gh,e^c=eh,f^c=fi,g^c=gh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 27819);