#495671

order	512
nilpotency class	2
isoclinism	isoclinic class of #392
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₁₆	C₂³
commutator, socle	C₂³	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#2388	C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1034 conjugacy classes of subgroups; only the 282 normal subgroups are shown.

normal subgroups only

This group has 1034 conjugacy classes of subgroups — too many to draw. Only its 282 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)(3,4)(5,6)(7,8)(9,10,13,18,14,19,26,32,15,12,17,24,27,25,31,37)(11,16,22,29,23,30,36,39,20,21,28,34,33,35,38,40);(2,4)(9,11)(10,16)(12,21)(13,22)(14,23)(15,20)(17,28)(18,29)(19,30)(24,34)(25,35)(26,36)(27,33)(31,38)(32,39)(37,40);(6,8)(11,20)(16,21)(22,28)(23,33)(29,34)(30,35)(36,38)(39,40);(9,12,13,24,14,25,26,37,15,10,17,18,27,19,31,32)(11,21,22,34,23,35,36,40,20,16,28,29,33,30,38,39)

Polycyclic presentation:

2^9|a^2=g,d^2=g,g^2=h,h^2=i,b^a=be,c^a=cf,c^b=ci

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402867);