#250829

order	512
nilpotency class	3
isoclinism	isoclinic class of #250775
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	#447	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

956 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 158025);