#499553

order	512
nilpotency class	4
isoclinism	isoclinic class of #498795
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 604 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:

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

Polycyclic presentation:

2^9|a^2=ei,b^2=ef,c^2=e,g^2=hi,h^2=i,b^a=be,c^a=cf,c^b=cg,g^b=gh,h^b=hi,d^c=di,g^c=gh,h^c=hi

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, 406749);