#358866

order	512
nilpotency class	3
isoclinism	isoclinic class of #23444
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#379
prime radical	#62686	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2518 conjugacy classes of subgroups; only the 370 normal subgroups are shown.

normal subgroups only

This group has 2518 conjugacy classes of subgroups — too many to draw. Only its 370 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 — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,d^2=g,h^2=i,b^a=be,c^a=cf,d^a=di,e^a=ei,c^b=cg,f^b=fi,h^b=hi,e^c=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72282010
maximal normal subgroup of	#72282010
maximal quotient of	#72282116

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 266062);