#316171

order	512
nilpotency class	3
isoclinism	isoclinic class of #316170
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

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

normal subgroups only

This group has 500 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) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=egh,b^2=e,c^2=ef,d^2=e,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,d^c=di,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, 223367);