#415668

order	512
nilpotency class	3
isoclinism	isoclinic class of #394328
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2617
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15204
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

875 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 322864);