#330719

order	512
nilpotency class	3
isoclinism	isoclinic class of #329743
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

512 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 138 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:

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

Polycyclic presentation:

2^9|a^2=efi,c^2=h,d^2=ei,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dhi,c^b=cg,d^b=dh,g^b=gi,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, 237915);