#316150

order	512
nilpotency class	3
isoclinism	isoclinic class of #316149
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2504
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#124
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

501 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 501 conjugacy classes of subgroups — too many to draw. Only its 158 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) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=e,c^2=ef,d^2=eh,e^2=h,f^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gh,c^b=cg,f^b=fh,g^c=gh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 223346);