#245995

order	512
nilpotency class	3
isoclinism	isoclinic class of #245903
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	C₂×#553	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1000 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 153191);