#479355

order	512
nilpotency class	3
isoclinism	isoclinic class of #478348
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2754
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#17201
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

704 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 386551);