#214770

order	512
nilpotency class	3
isoclinism	isoclinic class of #214768
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#40488	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1109 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=g,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=ce,d^b=di,f^b=fh,g^b=gh,f^c=fi,g^d=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, 121966);