#491296

order	512
nilpotency class	4
isoclinism	isoclinic class of #491227
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	#3147	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

420 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 126 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)(19,28,30,42)(20,25,22,37)(21,35,36,48)(24,38,39,32)(26,34,41,29)(31,43,33,47)(40,45,44,46);(1,2)(3,6)(4,8)(5,7)(9,13)(10,14)(11,15)(12,16)(17,19,29,45,22,33,36,38,23,30,34,46,20,31,21,32)(18,24,35,43,25,40,41,42,27,39,48,47,37,44,26,28);(1,3,5,10)(2,6,7,14)(4,12,11,9)(8,16,15,13)(19,30)(24,39)(28,42)(31,33)(32,38)(40,44)(43,47)(45,46);(2,7)(6,14)(8,15)(13,16)(17,20,23,22)(18,25,27,37)(19,31,30,33)(21,34,36,29)(24,40,39,44)(26,35,41,48)(28,43,42,47)(32,46,38,45)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398492);