#362594

order	512
nilpotency class	3
isoclinism	isoclinic class of #362481
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2293
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#13919
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

863 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)(3,11,17,25)(4,18,21,10)(5,22,16,9)(6,12)(7,14)(8,13)(15,29)(19,24)(20,28,23,27)(26,30)(31,32);(1,3,7,16,8,17,6,5)(2,9,12,11,13,22,14,25)(4,19,20,30,21,31,23,15)(10,26,28,24,18,29,27,32);(1,4)(2,10)(3,15)(5,19)(6,20)(7,23)(8,21)(9,24)(11,26)(12,27)(13,18)(14,28)(16,31)(17,30)(22,32)(25,29);(1,5,6,17,8,16,7,3)(2,11,14,9,13,25,12,22)(4,19,20,30,21,31,23,15)(10,26,28,24,18,29,27,32)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 269790);