#488529

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	C₂×#3191	C₂
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

873 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 873 conjugacy classes of subgroups — too many to draw. Only its 134 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)(17,18,23,26)(19,28,31,46)(20,25,22,37)(21,33,36,47)(24,39,41,30)(27,32,43,35)(29,45,38,44)(34,40,48,42);(1,3)(2,6)(4,9)(5,10)(7,12)(8,13)(11,16)(14,15)(17,19)(18,24)(20,29)(21,34)(22,38)(23,31)(25,40)(26,41)(27,44)(28,33)(30,35)(32,39)(36,48)(37,42)(43,45)(46,47);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,16)(13,15)(17,20,23,22)(18,25,26,37)(19,29,31,38)(21,32,36,35)(24,40,41,42)(27,33,43,47)(28,45,46,44)(30,34,39,48);(2,8)(6,13)(7,14)(12,15)(18,26)(24,41)(25,37)(27,43)(28,46)(33,47)(40,42)(44,45)

Polycyclic presentation:

2^9|a^2=i,c^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, 395725);