#273765

order	512
nilpotency class	3
isoclinism	isoclinic class of #273755
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	#62091	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

802 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)(3,10,12,15)(4,8)(5,9)(6,11,7,13)(14,16)(17,18,22,26,23,28,21,27)(19,29,31,24,32,40,38,41)(20,33,35,44,37,43,36,25)(30,45,34,48,47,39,46,42);(1,3)(2,6)(4,13)(5,12)(7,9)(8,15)(10,16)(11,14)(17,19,23,32)(18,24,28,41)(20,34,37,46)(21,38,22,31)(25,42,44,48)(26,40,27,29)(30,35,47,36)(33,45,43,39);(4,14)(8,16)(10,15)(11,13)(17,20,23,37)(18,25,28,44)(19,30,32,47)(21,35,22,36)(24,39,41,45)(26,43,27,33)(29,42,40,48)(31,46,38,34);(1,3)(2,7)(4,13)(5,12)(6,9)(8,10)(11,14)(15,16)(17,18)(19,24)(20,25)(21,27)(22,26)(23,28)(29,38)(30,39)(31,40)(32,41)(33,35)(34,42)(36,43)(37,44)(45,47)(46,48)

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=h,e^2=i,f^2=h,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=fh,g^c=gh,e^d=ei,f^d=fhi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 180961);