#511538

order	512
nilpotency class	4
isoclinism	isoclinic class of #509081
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

473 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 473 conjugacy classes of subgroups — too many to draw. Only its 140 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)(17,18,21,24)(19,28,31,25)(20,32,23,26)(22,27,34,38)(29,39,46,41)(30,40,37,42)(33,44,35,43)(36,48,47,45);(1,3,5,8)(2,6,7,4)(9,11)(10,13)(12,16)(14,15)(18,24)(19,29)(20,23)(22,35)(25,39)(27,44)(28,41)(30,47)(31,46)(33,34)(36,37)(38,43)(40,48)(42,45);(10,14)(13,15)(17,19,21,31)(18,25,24,28)(20,30,23,37)(22,36,34,47)(26,40,32,42)(27,45,38,48)(29,33,46,35)(39,43,41,44);(1,4,5,6)(2,3,7,8)(11,16)(13,15)(17,20,21,23)(18,26,24,32)(19,30,31,37)(22,33,34,35)(25,40,28,42)(27,43,38,44)(29,36,46,47)(39,45,41,48)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 418734);