#427949

order	512
nilpotency class	3
isoclinism	isoclinic class of #427949
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15376
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23400	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1283 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)(3,11,13,25)(4,15,16,8)(5,9)(6,10)(7,14,23,19)(12,27)(17,26,20,24)(18,21)(22,28)(29,32)(30,31);(1,2)(3,7)(4,15)(5,18)(6,10)(8,16)(9,21)(11,19)(12,22)(13,23)(14,25)(17,24)(20,26)(27,28)(29,31)(30,32);(1,3)(2,7)(4,12)(5,19)(6,13)(8,22)(9,25)(10,23)(11,18)(14,21)(15,31)(16,29)(17,30)(20,28)(24,27)(26,32);(1,4,5,20,6,16,21,17)(2,8,9,26,10,15,18,24)(3,12,19,28,13,29,14,30)(7,22,25,32,23,31,11,27)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 335145);