#214769

order	512
nilpotency class	3
isoclinism	isoclinic class of #214768
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

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	#40490	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1198 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)(3,15,19,26)(4,20,18,30)(5,14)(6,9)(7,13)(8,16,25,22)(10,21,29,31)(11,23)(12,17)(24,28)(27,32);(1,3)(2,8)(4,6)(5,16)(7,19)(9,10)(11,15)(12,30)(13,25)(14,26)(17,21)(18,24)(20,27)(22,23)(28,29)(31,32);(2,9)(3,16)(6,17)(8,20)(10,15)(11,27)(12,14)(13,28)(19,22)(24,32)(25,30)(26,29);(1,4,7,18)(2,10,13,29)(3,17,19,32)(5,21,23,31)(6,22,24,16)(8,12,25,27)(9,15,28,26)(11,20,14,30)

Polycyclic presentation:

2^9|a^2=g,d^2=h,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, 121965);