#263540

order	512
nilpotency class	3
isoclinism	isoclinic class of #263527
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#62328	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

845 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 845 conjugacy classes of subgroups — too many to draw. Only its 140 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)(18,23)(20,34)(21,24)(25,39)(26,42)(27,28)(29,41)(30,46)(31,32)(35,47)(36,38)(45,48);(4,13)(6,15)(7,12)(10,14)(17,18,24,28,22,27,21,23)(19,29,32,25,33,39,31,41)(20,26,38,44,36,42,34,37)(30,45,47,40,35,48,46,43);(1,3,5,11)(2,6,8,15)(4,14,13,10)(7,16,12,9)(17,19)(18,25)(20,35)(21,31)(22,33)(23,29)(24,32)(26,43)(27,41)(28,39)(30,36)(34,47)(37,48)(38,46)(40,42)(44,45);(17,20,24,38,22,36,21,34)(18,26,28,44,27,42,23,37)(19,30,32,47,33,35,31,46)(25,40,39,48,41,43,29,45)

Polycyclic presentation:

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

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, 170736);