#471157

order	512
nilpotency class	3
isoclinism	isoclinic class of #471157
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16765
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

670 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)(3,7,13,19)(4,15,16,28)(5,8)(6,10)(9,17,21,24)(11,22)(12,25)(14,20,23,31)(18,29)(26,32)(27,30);(1,3,11,23,5,13,6,14)(2,7,10,20,8,19,22,31)(4,12,24,29,16,26,17,27)(9,18,15,25,21,30,28,32);(2,8)(6,11)(12,26)(14,23)(15,28)(17,24)(20,31)(25,32);(1,4,5,16)(2,9)(3,12,13,26)(6,17,11,24)(7,18)(8,21)(10,15)(14,27,23,29)(19,30)(20,25)(22,28)(31,32)

Polycyclic presentation:

2^9|a^2=efg,b^2=efh,d^2=ef,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,d^c=de,h^c=hi,g^d=gi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 378353);