#248058

order	512
nilpotency class	3
isoclinism	isoclinic class of #247975
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³	C₂×#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

992 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=hi,d^2=hi,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gi,c^b=cg,e^b=eh,f^b=fi,d^c=di,f^c=fi,e^d=eh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 155254);