#215603

order	512
nilpotency class	3
isoclinism	isoclinic class of #195990
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator, prime radical	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

822 conjugacy classes of subgroups; only the 226 normal subgroups are shown.

normal subgroups only

This group has 822 conjugacy classes of subgroups — too many to draw. Only its 226 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) = 256

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

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

Polycyclic presentation:

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

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, 122799);