#250893

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

961 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)(18,24)(20,34)(21,22)(25,29)(27,28)(30,44)(31,32)(35,36)(37,41)(38,47)(39,40)(45,46);(1,3,8,15)(2,9,13,16)(4,12,10,11)(5,7,14,6)(17,18,23,28)(19,29,33,39)(20,26,35,42)(21,27,22,24)(25,31,40,32)(30,43,45,48)(34,37,36,41)(38,46,47,44);(1,4,8,10)(2,5,13,14)(3,12,15,11)(6,9,7,16)(17,19,23,33)(18,25,28,40)(20,30,35,45)(21,32,22,31)(24,29,27,39)(26,38,42,47)(34,46,36,44)(37,43,41,48);(1,5)(2,10)(3,7)(4,13)(6,15)(8,14)(9,11)(12,16)(17,20,23,35)(18,26,28,42)(19,30,33,45)(21,36,22,34)(24,37,27,41)(25,38,40,47)(29,43,39,48)(31,46,32,44)

Polycyclic presentation:

2^9|a^2=hi,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, 158089);