#496502

order	512
nilpotency class	4
isoclinism	isoclinic class of #496502
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#21865
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#60710	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2873 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

764 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=e,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=deh,e^a=ei,c^b=ce,f^b=fi,g^b=gi,f^c=fh,g^c=gh,h^c=hi,f^d=fi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403698);