#118364

order	512
nilpotency class	3
isoclinism	isoclinic class of #118362
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

813 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=h,e^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=fi,d^c=di,e^c=eh,g^c=ghi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 25560);