#299277

order	512
nilpotency class	3
isoclinism	isoclinic class of #299253
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

974 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 974 conjugacy classes of subgroups — too many to draw. Only its 174 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 206473);