#99305

order	512
nilpotency class	3
isoclinism	isoclinic class of #10304
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#127
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
socle, prime radical	C₂³	#384
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

306 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dgh,c^2=eg,d^2=g,e^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,c^b=ci

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6501);