#325969

order	512
nilpotency class	3
isoclinism	isoclinic class of #288715
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2537	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

634 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 634 conjugacy classes of subgroups — too many to draw. Only its 146 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)(3,8)(4,6)(5,7)(9,10)(11,17)(12,16)(13,20)(14,19)(15,21)(18,23)(22,24)(25,26)(27,30)(28,31)(29,32)(33,40)(34,38)(35,37)(36,39);(1,3,5,4)(2,6,7,8)(9,11,14,22,15,23,13,12)(10,16,20,18,21,24,19,17)(25,27,29,35,28,34,36,33)(26,30,32,37,31,38,39,40);(1,4,5,3)(2,6,7,8)(9,12,13,23,15,22,14,11)(10,17,19,24,21,18,20,16)(26,31)(27,33)(29,36)(30,37)(34,35)(38,40);(2,7)(6,8)(9,12,13,23,15,22,14,11)(10,18,19,16,21,17,20,24)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 233165);