#491501

order	512
nilpotency class	4
isoclinism	isoclinic class of #491437
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#493
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂³	#493
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

517 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 517 conjugacy classes of subgroups — too many to draw. Only its 162 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)(3,10)(4,8)(5,14)(6,9)(7,12)(11,15)(13,16)(17,18,22,26)(19,27,28,41)(20,32,34,47)(21,35,33,25)(23,36,37,30)(24,39,40,29)(31,46,44,43)(38,45,42,48);(1,3,5,12,6,13,4,11)(2,7,8,10,9,15,14,16)(19,28)(23,37)(27,41)(30,36)(31,44)(38,42)(43,46)(45,48);(17,19,29,45,21,31,34,36,22,28,39,48,33,44,20,30)(18,23,32,43,25,38,40,41,26,37,47,46,35,42,24,27);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,10,15,16)(19,28)(23,37)(27,41)(30,36)(31,44)(38,42)(43,46)(45,48)

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=fh,d^2=g,e^2=g,f^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fh,h^a=hi,c^b=ci,d^c=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225180, #72225191, #72225195, …
maximal normal subgroup of	#72225180, #72225191, #72225195, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398697);