#219270

order	512
nilpotency class	3
isoclinism	isoclinic class of #23924
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle, prime radical	C₂³	C₂×C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ #1744 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

508 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 186 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) = 512

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=efh,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ce,d^b=dh,d^c=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 126466);