#484104

order	512
nilpotency class	4
isoclinism	isoclinic class of #484095
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#442
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂×#135	C₂×C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1451 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

435 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=fg,d^2=i,e^2=g,f^2=h,g^2=i,b^a=be,c^a=cg,d^a=dhi,e^a=eg,g^a=gi,e^b=egi,f^b=fi,g^b=gi,d^c=dh,e^c=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72216702, #72216909, #72216945, …
maximal normal subgroup of	#72216702, #72216909, #72216945, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391300);