#270761

order	512
nilpotency class	3
isoclinism	isoclinic class of #270747
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	#3480	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

812 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=hi,d^2=i,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dh,e^b=eh,g^b=gi,f^c=fh,g^c=gi,e^d=eh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 177957);