#249351

order	512
nilpotency class	3
isoclinism	isoclinic class of #249319
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3006	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1208 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=hi,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fhi,e^c=eh,f^c=fi,f^d=fh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 156547);