#118958

order	512
nilpotency class	3
isoclinism	isoclinic class of #118955
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1440
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

761 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:

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

Polycyclic presentation:

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

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, 26154);