#214638

order	512
nilpotency class	3
isoclinism	isoclinic class of #214040
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	#37296	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

999 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)(19,25,20,23)(21,24)(22,26)(27,30)(28,32,29,31);(1,3)(2,6)(4,9)(5,10)(7,13)(8,14)(11,15)(12,16)(17,19)(18,23)(20,22)(21,29)(24,32)(25,26)(27,31)(28,30);(1,4)(2,7)(3,9)(5,12)(6,13)(8,11)(10,16)(14,15)(18,24)(21,30)(23,31)(25,32)(26,27)(28,29);(3,10)(6,14)(9,16)(13,15)(17,20,22,19)(18,25,26,23)(21,29,30,28)(24,32,27,31)

Polycyclic presentation:

2^9|a^2=g,d^2=h,f^2=h,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=fh,g^c=ghi,g^d=gh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 121834);