#272041

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×H(5,𝔽₂)	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

874 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)(19,24)(20,32)(21,23)(22,27)(25,33)(26,36)(28,39)(29,44)(30,37)(31,40)(34,41)(35,42)(38,43)(45,47)(46,48);(18,23)(19,28)(21,36)(25,32)(26,27)(29,43)(30,31)(33,34)(37,39)(41,42)(44,47)(45,46);(1,3,6,12)(2,7,9,16)(4,11,5,14)(8,15,13,10)(17,19,22,31)(18,24,27,40)(20,29,35,45)(21,30,36,28)(23,37,26,39)(25,38,41,48)(32,44,42,47)(33,43,34,46);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(17,20)(18,25)(19,29)(21,34)(22,35)(23,32)(24,38)(26,42)(27,41)(28,43)(30,46)(31,45)(33,36)(37,44)(39,47)(40,48)

Polycyclic presentation:

2^9|a^2=f,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, 179237);