#511524

order	512
nilpotency class	4
isoclinism	isoclinic class of #509065
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

468 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 138 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=ef,c^2=i,d^2=efi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,g^b=gh,h^b=hi,g^c=gh,h^c=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 418720);