#405853

order	512
nilpotency class	3
isoclinism	isoclinic class of #405853
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14919
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1045 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 313049);