#415932

order	512
nilpotency class	3
isoclinism	isoclinic class of #393944
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×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

1706 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)(11,15)(12,13)(14,16)(17,18,22,27)(19,28,30,24)(20,26,21,25)(23,31,32,29);(1,2,3,4)(10,13)(14,15)(17,19,20,31,22,30,21,29)(18,23,25,24,27,32,26,28);(6,8)(17,19,22,30)(18,24,27,28)(20,29,21,31)(23,26,32,25);(9,11,12,16)(10,14,13,15)(17,20,22,21)(18,25,27,26)(19,29,30,31)(23,28,32,24)

Polycyclic presentation:

2^9|a^2=i,b^2=eh,c^2=i,d^2=h,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=di,e^c=ei,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 323128);