#284977

order	512
nilpotency class	3
isoclinism	isoclinic class of #284923
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3511	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

653 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2,7,9,6,4,5,10)(3,11,13,8,15,16,14,12)(17,18,21,31,22,27,23,26)(19,25,30,32,28,24,20,29);(1,3,7,13,6,15,5,14)(2,8,9,16,4,12,10,11)(17,19,23,20,22,28,21,30)(18,24,26,32,27,25,31,29);(1,2,6,4)(3,12,15,8)(5,9,7,10)(11,13,16,14)(17,20)(18,25)(19,23)(21,28)(22,30)(24,27)(26,32)(29,31);(1,4,6,2)(3,8,15,12)(5,9,7,10)(11,13,16,14)(17,21,22,23)(18,26,27,31)(19,20,28,30)(24,29,25,32)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 192173);