#250413

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	#62503	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

989 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=h,d^2=i,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, 157609);