#213804

order	512
nilpotency class	3
isoclinism	isoclinic class of #213798
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	C₂×C₄²	C₂⁴
prime radical	#35534	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

938 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)(3,10,12,6)(4,9)(5,8)(7,11,16,13)(14,15)(17,18,21,26,22,25,19,23)(20,24,28,31,29,32,27,30);(1,3)(2,6)(4,13)(5,12)(7,15)(8,10)(9,16)(11,14)(17,19,22,21)(18,23,25,26)(20,28,29,27)(24,31,32,30);(1,3)(2,7)(4,13)(5,12)(6,9)(8,16)(10,15)(11,14)(17,20)(18,24)(19,27)(21,28)(22,29)(23,30)(25,32)(26,31);(1,3)(2,6)(4,11)(5,12)(7,9)(8,10)(13,14)(15,16)(18,25)(20,29)(23,26)(27,28)

Polycyclic presentation:

2^9|a^2=g,b^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cei,f^b=fh,g^b=gh,d^c=di,f^c=fh,g^c=gh,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, 121000);