#280980

order	512
nilpotency class	3
isoclinism	isoclinic class of #280955
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2420
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³	#546
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

431 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)(17,18,24,29)(19,30,33,27)(20,34,38,43)(21,25,32,41)(22,39,23,28)(26,36,44,37)(31,40,47,48)(35,42,46,45);(2,6)(3,9)(5,11)(7,15)(10,12)(13,16)(17,19,22,21,24,33,23,32)(18,25,28,30,29,41,39,27)(20,35,37,31,38,46,36,47)(26,42,34,48,44,45,43,40);(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)(23,37)(24,38)(25,40)(27,42)(28,43)(29,44)(30,45)(32,46)(33,47)(34,39)(41,48);(2,6)(3,10)(8,14)(9,12)(17,21,22,33,24,32,23,19)(18,27,28,25,29,30,39,41)(20,35,36,47,38,46,37,31)(26,42,43,40,44,45,34,48)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 188176);