#404974

order	512
nilpotency class	3
isoclinism	isoclinic class of #393909
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

1234 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)(9,10,13,12)(11,14,15,16)(17,18)(19,27)(20,23)(21,26)(22,25)(24,29)(28,32)(30,31);(1,2,3,4)(9,11)(10,14)(12,16)(13,15)(18,23)(24,31)(25,26)(27,32);(6,8)(11,15)(14,16)(17,19,20,28)(18,24,23,31)(21,30,22,29)(25,32,26,27);(9,12,13,10)(11,16,15,14)(18,25)(21,22)(23,26)(24,32)(27,31)(29,30)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 312170);