#277942

order	512
nilpotency class	3
isoclinism	isoclinic class of #277931
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#532	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

526 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)(3,6)(4,7)(5,8)(9,16)(10,14)(11,13)(12,15)(17,18,24,28,23,30,22,29)(19,31,40,27,21,39,33,25)(20,34,36,26,38,44,37,43)(32,42,46,45,35,41,47,48);(1,3,4,10)(2,6,7,14)(5,9,12,11)(8,16,15,13)(17,19,22,33,23,21,24,40)(18,25,28,31,30,27,29,39)(20,32,36,46,38,35,37,47)(26,41,34,45,43,42,44,48);(2,7)(3,9)(5,12)(6,13)(10,11)(14,16)(17,20)(18,26)(19,32)(21,35)(22,36)(23,38)(24,37)(25,41)(27,42)(28,34)(29,44)(30,43)(31,45)(33,46)(39,48)(40,47);(17,21,24,33,23,19,22,40)(18,27,29,31,30,25,28,39)(20,35,37,46,38,32,36,47)(26,42,44,45,43,41,34,48)

Polycyclic presentation:

2^9|a^2=fhi,b^2=ei,d^2=eh,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,g^b=gi,f^c=fh,g^c=gi

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, 185138);