#409675

order	512
nilpotency class	3
isoclinism	isoclinic class of #395151
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×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 ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

852 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 316871);