#289078

order	512
nilpotency class	3
isoclinism	isoclinic class of #289075
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3036	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

625 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:

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

Polycyclic presentation:

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

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, 196274);