#481318

order	512
nilpotency class	4
isoclinism	isoclinic class of #11651
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂²×#102
prime radical	#60673	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

774 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)(19,28,31,45)(20,24)(21,35)(22,27)(23,30,40,47)(25,34)(26,33)(29,44,32,42)(36,39,48,38)(37,41)(43,46);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,19)(18,23)(20,29)(21,36)(22,31)(24,38)(25,39)(26,42)(27,40)(28,41)(30,46)(32,34)(33,47)(35,45)(37,48)(43,44);(3,9)(6,12)(10,15)(13,16)(17,20,22,34)(18,24,27,25)(19,29,31,32)(21,33,37,46)(23,38,40,39)(26,41,43,35)(28,44,45,42)(30,36,47,48);(17,20,22,34)(18,25,27,24)(19,29,31,32)(21,33,37,46)(23,39,40,38)(26,35,43,41)(28,42,45,44)(30,36,47,48)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286868, #72287111, #72287143, …
maximal normal subgroup of	#72286868, #72287111, #72287143, …
maximal quotient of	#72343807

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 388514);