#396612

order	512
nilpotency class	3
isoclinism	isoclinic class of #396612
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2850
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14735
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1139 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)(3,13)(4,9)(5,15)(6,24)(7,12)(8,11)(10,23)(14,29)(16,22)(17,18)(19,31)(20,27)(21,28)(25,26)(30,32);(1,3,7,10)(2,5,12,22)(4,17)(6,16,8,15)(9,25)(11,23,24,13)(14,19)(18,31)(20,26)(21,30)(27,32)(28,29);(1,4,6,19,7,21,8,20)(2,9,11,27,12,28,24,31)(3,14,15,17,10,26,16,30)(5,18,13,29,22,32,23,25);(1,5)(2,10)(3,12)(4,18)(6,13)(7,22)(8,23)(9,26)(11,16)(14,28)(15,24)(17,31)(19,29)(20,25)(21,32)(27,30)

Polycyclic presentation:

2^9|b^2=f,c^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,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, 303808);