#498768

order	512
nilpotency class	2
isoclinism	isoclinic class of #2812
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₈	C₂⁴
commutator, socle, prime radical	C₂³	C₂³×C₈
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

428 conjugacy classes of subgroups; only the 214 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 214 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) = 512

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=ef,d^2=e,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cefi,d^b=dei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405964);