#480720

order	512
nilpotency class	4
isoclinism	isoclinic class of #480662
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁴
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

988 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 988 conjugacy classes of subgroups — too many to draw. Only its 202 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=hi,d^2=i,f^2=h,b^a=be,d^a=dh,e^a=eg,g^a=gi,c^b=chi,d^b=di,f^b=fg,f^e=fi

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, 387916);