#424166

order	512
nilpotency class	3
isoclinism	isoclinic class of #417589
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1434 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 331362);