#239075

order	512
nilpotency class	3
isoclinism	isoclinic class of #233220
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1182 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 1182 conjugacy classes of subgroups — too many to draw. Only its 210 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) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

(1,2)(3,4)(6,10)(7,8)(11,12)(13,14,18,22)(15,23,26,38)(16,27,30,20)(17,31,33,21)(19,32,35,25)(24,40,28,39)(29,37,44,42)(34,41,36,43);(6,11)(7,8)(10,12)(13,15,18,26)(14,19,22,35)(16,28,30,24)(17,32,33,25)(20,36,27,34)(21,38,31,23)(29,41,44,43)(37,39,42,40);(2,4)(5,6)(7,11)(8,10)(9,12)(13,16,17,29,18,30,33,44)(14,20,21,37,22,27,31,42)(15,24,25,41,26,28,32,43)(19,34,23,39,35,36,38,40);(6,12)(10,11)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=e,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=gi,c^b=cg,e^b=eh,f^b=fi,g^b=gi,d^c=di,f^c=fhi,g^c=ghi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 146271);