#273976

order	512
nilpotency class	3
isoclinism	isoclinic class of #273947
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	#3499	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

846 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

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

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, 181172);