#483326

order	512
nilpotency class	4
isoclinism	isoclinic class of #33404
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#106
commutator	C₈	C₂³×C₈
Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂×#41	C₄⋊C₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₄×#131 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

514 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,e^2=g,f^2=h,g^2=i,b^a=be,e^a=eg,g^a=gi,f^b=fi,d^c=di

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72216392, #72216441, #72216456, …
maximal normal subgroup of	#72216392, #72216441, #72216456, …
maximal quotient of	D₄×#1368, #72231928, #72231931, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390522);