#118421

order	512
nilpotency class	3
isoclinism	isoclinic class of #118417
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

807 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=h,e^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,c^b=cf,f^b=fhi,d^c=di,e^c=eh,f^c=fh,g^c=ghi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 25617);