#263325

order	512
nilpotency class	3
isoclinism	isoclinic class of #263311
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	#41
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

823 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 823 conjugacy classes of subgroups — too many to draw. Only its 146 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)(5,8)(6,10)(7,13)(11,16)(14,15)(17,18,22,26)(19,27,31,41)(20,32,33,46)(21,35,28,25)(23,36,37,30)(24,39,40,34)(29,43,44,38)(42,47,48,45);(4,13)(6,15)(7,12)(10,14)(17,19,28,36,22,31,21,30)(18,23,35,41,26,37,25,27)(20,29,39,47,33,44,34,45)(24,38,46,48,40,43,32,42);(1,3,5,11)(2,6,8,15)(4,14,13,10)(7,16,12,9)(20,33)(23,37)(24,40)(27,41)(29,44)(32,46)(34,39)(45,47);(17,20,28,39,22,33,21,34)(18,24,35,46,26,40,25,32)(19,29,36,47,31,44,30,45)(23,38,41,48,37,43,27,42)

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=i,d^2=eh,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,f^b=fhi,d^c=dh,f^c=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 170521);