#392351

order	512
nilpotency class	3
isoclinism	isoclinic class of #392203
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2280
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14683
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1219 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)(3,13,17,22)(4,18,21,25)(5,8)(6,11)(7,12)(9,16,24,15)(10,19,26,20)(14,23)(27,29)(28,31)(30,32);(2,8,12,11)(9,22,24,13)(10,18,26,25)(23,31,32,27);(1,3,5,15,7,17,6,16)(2,9,11,13,12,24,8,22)(4,14,19,28,21,30,20,29)(10,23,25,27,26,32,18,31);(1,4)(2,10)(3,14)(5,19)(6,20)(7,21)(8,18)(9,23)(11,25)(12,26)(13,27)(15,28)(16,29)(17,30)(22,31)(24,32)

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,h^c=hi,g^d=gi,h^d=hi

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, 299547);