#467694

order	512
nilpotency class	3
isoclinism	isoclinic class of #467694
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16584
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

696 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:

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

Polycyclic presentation:

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

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, 374890);