#469731

order	512
nilpotency class	3
isoclinism	isoclinic class of #469731
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16661
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

977 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)(3,12)(4,17,19,31)(5,22)(6,10)(7,11)(8,15)(9,21,26,20)(13,27)(14,25,18,23)(16,24)(28,30,32,29);(3,13)(5,6)(10,22)(12,27)(14,18)(17,31)(20,21)(28,32);(1,3,5,15,7,13,6,16)(2,8,10,12,11,24,22,27)(4,18,20,30,19,14,21,29)(9,25,17,32,26,23,31,28);(1,4)(2,9,11,26)(3,14,13,18)(5,20)(6,21)(7,19)(8,23)(10,17,22,31)(12,28)(15,29,16,30)(24,25)(27,32)

Polycyclic presentation:

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

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, 376927);