#489829

order	512
nilpotency class	4
isoclinism	isoclinic class of #489701
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	C₄×C₈	C₂⁴
prime radical	C₂²×D₈	D₄
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

716 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 136 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=hi,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dhi,e^a=eg,f^a=fh,g^a=gi,e^b=eg,f^b=fh,g^b=gi,e^c=eh,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, 397025);