#397205

order	512
nilpotency class	3
isoclinism	isoclinic class of #396612
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2850
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14747
prime radical	#60469	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1452 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=f,c^2=e,g^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,e^b=ei,f^c=fi,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 304401);