#484017

order	512
nilpotency class	4
isoclinism	isoclinic class of #33830
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂×C₄×D₈
prime radical	#2184	C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

444 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72234320, #72260298, #72260357, …
maximal normal subgroup of	#72234320, #72260298, #72260357, …
maximal quotient of	#72320165, #72320199

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391213);