#400141

order	512
nilpotency class	3
isoclinism	isoclinic class of #393940
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

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

1592 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)(5,6)(7,8)(9,10,12,13)(11,14,15,16)(17,18,21,26,23,20,22,25)(19,27,30,32,31,28,29,24);(2,4)(10,13)(14,16)(17,19,23,31)(18,24,20,32)(21,29,22,30)(25,27,26,28);(2,4)(17,20,23,18)(19,28,31,27)(21,26,22,25)(24,30,32,29);(6,8)(9,11,12,15)(10,14,13,16)(17,21,23,22)(18,25,20,26)(19,29,31,30)(24,27,32,28)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 307337);