#250412

order	512
nilpotency class	3
isoclinism	isoclinic class of #250327
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	#62090	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

978 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)(3,11)(4,9)(5,16)(6,13)(7,12)(8,14)(10,15)(17,18,23,19)(20,24,32,27)(21,25,34,28)(22,29,37,26)(30,36,39,33)(31,41,44,38)(35,42,46,40)(43,45,48,47);(1,3)(2,9)(4,7)(5,6)(8,15)(10,12)(11,13)(14,16)(17,19,23,18)(20,30,32,39)(21,28,34,25)(22,26,37,29)(24,33,27,36)(31,43,44,48)(35,40,46,42)(38,45,41,47);(1,4,8,5)(2,10,14,11)(3,6,15,7)(9,13,16,12)(17,20)(18,24)(19,27)(21,31)(22,36)(23,32)(25,38)(26,30)(28,41)(29,39)(33,37)(34,44)(35,47)(40,43)(42,48)(45,46);(1,5,8,4)(2,11,14,10)(3,6,15,7)(9,13,16,12)(17,21)(18,25)(19,28)(20,31)(22,35)(23,34)(24,38)(26,40)(27,41)(29,42)(30,43)(32,44)(33,45)(36,47)(37,46)(39,48)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=h,d^2=h,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,g^a=gi,c^b=cg,e^b=eh,g^b=gi,e^c=ei,f^c=fh,g^c=gi,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, 157608);