#331425

order	512
nilpotency class	3
isoclinism	isoclinic class of #331190
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄²×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2867	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

816 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 816 conjugacy classes of subgroups — too many to draw. Only its 170 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)(9,10,14,16,13,18,12,17)(11,15,21,22,19,23,20,24)(25,26)(27,30)(28,31)(29,32)(33,37)(34,40)(35,39)(36,38);(2,5)(6,8)(9,11)(10,15)(12,20)(13,19)(14,21)(16,22)(17,24)(18,23)(25,27,33,34,28,35,29,36)(26,30,37,40,31,39,32,38);(3,7)(6,8)(10,16)(12,14)(15,22)(17,18)(20,21)(23,24)(26,31)(27,34)(29,33)(30,38)(35,36)(39,40);(1,3,4,7)(2,6,5,8)(11,19)(15,23)(20,21)(22,24)

Polycyclic presentation:

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

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, 238621);