#481066

order	512
nilpotency class	4
isoclinism	isoclinic class of #33388
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#102
commutator	C₈	C₂³×C₈
Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂×#576	C₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₄×#131 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1037 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72216438, #72216441, #72216448, …
maximal normal subgroup of	#72216438, #72216441, #72216448, …
maximal quotient of	D₄×#1333, C₂×#481066, #72287052, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 388262);