#95226

order	512
nilpotency class	3
isoclinism	isoclinic class of #7035
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#95
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₄³	C₂³
socle	C₂³	C₂×#95
prime radical	C₂⁴	#95
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

528 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 174 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 — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=gi,d^2=gi,e^2=h,f^2=g,b^a=bd,d^a=dg,f^a=fghi,d^b=dh,e^b=ei

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, 2422);