#316153

order	512
nilpotency class	3
isoclinism	isoclinic class of #316149
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2504
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#33179	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 605 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=eghi,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, 223349);