#481390

order	512
nilpotency class	4
isoclinism	isoclinic class of #481373
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂⁴	#102
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

744 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,4,8,5,7,11,14)(3,6,10,13,9,12,15,16)(17,18)(19,27,30,43)(20,32)(21,35)(22,25)(23,29,39,45)(24,34)(26,33)(28,44,31,42)(36,38,46,40)(37,47)(41,48);(1,3)(2,6)(4,10)(5,9)(7,12)(8,13)(11,15)(14,16)(17,19,22,30)(18,23,25,39)(20,28,34,31)(21,36,37,46)(24,38,32,40)(26,42,41,44)(27,35,43,47)(29,33,45,48);(3,9)(6,12)(10,15)(13,16)(17,20,22,34)(18,24,25,32)(19,28,30,31)(21,33,37,48)(23,38,39,40)(26,35,41,47)(27,42,43,44)(29,36,45,46);(2,7)(6,12)(8,14)(13,16)(18,25)(23,39)(24,32)(26,41)(27,43)(35,47)(38,40)(42,44)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318517, #72318559, #72318568, …
maximal normal subgroup of	#72318517, #72318559, #72318568, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 388586);