#400115

order	512
nilpotency class	3
isoclinism	isoclinic class of #393928
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2433
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14796
prime radical	#23434	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1395 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:

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

Polycyclic presentation:

2^9|a^2=e,d^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,g^c=gi,h^c=hi,e^d=ei,f^d=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, 307311);