#491306

order	512
nilpotency class	4
isoclinism	isoclinic class of #491239
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#3169	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

417 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 417 conjugacy classes of subgroups — too many to draw. Only its 128 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398502);