#250427

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	#3514	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

952 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)(3,13,5,16)(4,15,9,14)(6,11,7,10)(17,18,22,26)(19,23,29,37)(20,24,30,38)(21,32,33,25)(27,40,34,35)(28,42,43,36)(31,45,44,39)(41,47,46,48);(1,3)(2,4)(5,8)(6,15)(7,14)(9,12)(10,16)(11,13)(19,27)(21,33)(23,35)(25,32)(28,41)(29,34)(31,44)(36,47)(37,40)(39,45)(42,48)(43,46);(1,4,8,9)(2,5,12,3)(6,13,7,16)(10,15,11,14)(17,19)(18,23)(20,28)(21,34)(22,29)(24,36)(25,40)(26,37)(27,33)(30,43)(31,46)(32,35)(38,42)(39,48)(41,44)(45,47);(1,5)(2,9)(3,8)(4,12)(6,14)(7,15)(10,13)(11,16)(17,20)(18,24)(19,28)(21,31)(22,30)(23,36)(25,39)(26,38)(27,41)(29,43)(32,45)(33,44)(34,46)(35,47)(37,42)(40,48)

Polycyclic presentation:

2^9|a^2=hi,c^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, 157623);