#184603

order	512
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂²×C₈

Aliases

No aliases for this group.

Subgroups

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

704 conjugacy classes of subgroups; only the 310 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 310 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)(3,7)(4,6)(5,8)(9,10,14,17,13,12,16,21)(11,18,20,24,19,15,22,23)(27,33)(30,34)(35,37)(38,40);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,18)(13,19)(14,20)(16,22)(17,23)(21,24)(25,26,28,31,29,32,36,39)(27,34,35,40,33,30,37,38);(3,8)(5,7)(9,12,14,21,13,10,16,17)(11,18,20,24,19,15,22,23)(25,27)(26,30)(28,35)(29,33)(31,38)(32,34)(36,37)(39,40);(2,6)(5,7)(10,12)(11,19)(17,21)(20,22)(27,33)(30,34)(35,37)(38,40)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,f^2=h,g^2=i,b^a=be,c^a=ci,d^a=de,c^b=cehi,d^b=dh,d^c=dhi

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 91799);