#191349

order	512
nilpotency class	3
isoclinism	isoclinic class of #8603
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2283
prime radical	C₂⁴	#117
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

449 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)(19,28,31,24)(20,25)(21,26)(22,27)(23,30,36,44)(29,41,43,37)(32,38)(33,39)(34,40)(35,42,46,48)(45,47);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,19)(18,23,27,36)(20,29)(21,30)(22,31)(24,26,28,40)(25,35,39,46)(32,42)(33,43)(34,44)(37,38,41,47)(45,48);(3,9)(6,12)(10,15)(13,16)(17,19)(18,24)(20,29)(21,30)(22,31)(23,26)(25,37)(27,28)(32,42)(33,43)(34,44)(35,38)(36,40)(39,41)(45,48)(46,47);(17,20,21,32,22,33,34,45)(18,25,26,38,27,39,40,47)(19,29,30,42,31,43,44,48)(23,35,28,41,36,46,24,37)

Polycyclic presentation:

2^9|a^2=g,b^2=f,d^2=e,e^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,c^b=ci,g^b=gh,g^c=gh

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, 98545);