#346497

order	512
nilpotency class	3
isoclinism	isoclinic class of #346482
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#62876	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2421 conjugacy classes of subgroups; only the 188 normal subgroups are shown.

normal subgroups only

This group has 2421 conjugacy classes of subgroups — too many to draw. Only its 188 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=hi,d^2=h,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fh,e^b=eh,f^b=fi,g^b=gh,e^c=ei,f^c=fh,e^d=eh

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 253693);