#488528

order	512
nilpotency class	4
isoclinism	isoclinic class of #488523
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁵	D₈
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 863 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,e^2=gi,g^2=i,b^a=be,c^a=cf,d^a=dhi,e^a=eg,g^a=gi,e^b=eg,f^b=fh,g^b=gi,e^c=eh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 395724);