#241630

order	512
nilpotency class	3
isoclinism	isoclinic class of #233789
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

702 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 702 conjugacy classes of subgroups — too many to draw. Only its 144 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) = 512

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=h,d^2=e,f^2=h,g^2=hi,b^a=be,c^a=cf,f^a=fh,g^a=gi,c^b=cg,d^b=dh,f^b=fi,g^b=ghi,d^c=dh,f^c=fh,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, 148826);