#299311

order	512
nilpotency class	3
isoclinism	isoclinic class of #299301
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
prime radical	#3038	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 942 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ei,c^2=f,d^2=eh,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,d^c=dh,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, 206507);