#400118

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	#23414	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1429 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)(3,14,16,9)(4,19)(5,18,24,17)(7,12,8,13)(10,25)(15,30)(20,29)(21,32)(22,28)(23,27)(26,31);(1,3)(2,9)(4,20)(5,13)(6,16)(7,17)(8,18)(10,21)(11,14)(12,24)(15,22)(19,29)(23,31)(25,32)(26,27)(28,30);(1,4)(2,10)(3,15)(5,21)(6,23)(7,25)(8,22)(9,26)(11,28)(12,27)(13,19)(14,29)(16,32)(17,20)(18,31)(24,30);(1,5,8,14,6,24,7,9)(2,3,13,18,11,16,12,17)(4,21,22,29,23,30,25,26)(10,15,19,31,28,32,27,20)

Polycyclic presentation:

2^9|a^2=e,d^2=ef,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,g^b=gi,e^c=ei,f^c=fi,g^c=gi,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, 307314);