#331472

order	512
nilpotency class	3
isoclinism	isoclinic class of #331233
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

695 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 695 conjugacy classes of subgroups — too many to draw. Only its 154 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=fh,b^2=ghi,c^2=i,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,d^b=dh,d^c=de,f^c=fh,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 238668);