#483423

order	512
nilpotency class	4
isoclinism	isoclinic class of #12028
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

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

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=ch,d^b=di,f^b=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287671, #72287698, #72287744, …
maximal normal subgroup of	#72287671, #72287698, #72287744, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390619);