#464522

order	512
nilpotency class	3
isoclinism	isoclinic class of #453981
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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

1156 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:

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

Polycyclic presentation:

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

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, 371718);