#272057

order	512
nilpotency class	3
isoclinism	isoclinic class of #272027
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×2₋¹⁺⁴	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

853 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=hi,d^2=i,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,e^b=eh,g^b=gh,f^c=fi,g^c=gh,e^d=eh

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 179253);