#483420

order	512
nilpotency class	4
isoclinism	isoclinic class of #12028
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#106
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂²×#106
prime radical	C₂⁴	#106
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

453 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)(19,28,29,41)(20,25)(21,34)(22,37)(23,27)(24,31,38,43)(26,35)(30,42,32,46)(33,40)(36,47)(39,44,48,45);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,19,21,31,22,32,35,45,23,29,36,43,20,30,33,44)(18,24,26,28,25,39,34,42,27,38,40,41,37,48,47,46);(3,9)(6,12)(10,15)(13,16)(19,29)(24,38)(28,41)(30,32)(31,43)(39,48)(42,46)(44,45);(17,20,23,22)(18,25,27,37)(19,30,29,32)(21,33,36,35)(24,39,38,48)(26,34,40,47)(28,42,41,46)(31,44,43,45)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285455, #72285482, #72285490, …
maximal normal subgroup of	#72285455, #72285482, #72285490, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390616);