#214628

order	512
nilpotency class	3
isoclinism	isoclinic class of #214020
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	C₂×#537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

970 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=ce,f^b=fh,g^b=gh,d^c=di,f^c=fh,g^c=ghi,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, 121824);