#93401

order	512
nilpotency class	4
isoclinism	isoclinic class of #1343
automorphism	?
homology	Abelianization (H₁) = C₄×C₈ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	#99
commutator	C₂²×C₄	C₄×C₈
Frattini, cosocle	C₂×C₄×M₁₆	C₂²
socle	C₂²	#1280
prime radical	C₂⁴	C₄⋊C₈
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂×C₄×M₁₆ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

273 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 273 conjugacy classes of subgroups — too many to draw. Only its 78 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 — 2 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=f,d^2=fi,e^2=h,h^2=i,b^a=bc,c^a=cf,e^a=efg,c^b=cg,g^b=gi,e^c=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 597);