#277927

order	512
nilpotency class	3
isoclinism	isoclinic class of #277891
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	C₂²×D₄	#41
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

554 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)(3,7)(4,6)(5,8)(9,16)(10,13)(11,14)(12,15)(17,18,24,28,23,30,22,29)(19,27,21,39,33,41,32,25)(20,34,36,26,38,44,37,43)(31,45,46,42,47,40,35,48);(2,6)(3,9)(5,11)(7,15)(10,12)(13,16)(17,19,22,32,23,33,24,21)(18,25,28,27,30,39,29,41)(20,31,36,46,38,47,37,35)(26,40,34,42,43,45,44,48);(1,3,4,10)(2,7,6,13)(5,12,11,9)(8,15,14,16)(17,20)(18,26)(19,31)(21,35)(22,36)(23,38)(24,37)(25,40)(27,42)(28,34)(29,44)(30,43)(32,46)(33,47)(39,45)(41,48);(2,6)(3,10)(8,14)(9,12)(17,21,24,33,23,32,22,19)(18,27,29,25,30,41,28,39)(20,35,37,47,38,46,36,31)(26,42,44,40,43,48,34,45)

Polycyclic presentation:

2^9|a^2=fh,b^2=ei,c^2=i,d^2=ehi,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,g^b=gi,d^c=di,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, 185123);