#414031

order	512
nilpotency class	3
isoclinism	isoclinic class of #400688
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2544
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15160
prime radical	#23414	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1243 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)(3,14)(4,9)(5,8)(6,11)(7,12)(10,21)(13,22)(15,24)(16,25)(17,26)(18,31)(19,28)(20,30)(23,29)(27,32);(1,3,6,5)(2,8,11,14)(4,17)(7,16,22,21)(9,26)(10,13,25,12)(15,23)(18,19)(20,32)(24,29)(27,30)(28,31);(1,4)(2,9)(3,15)(5,18)(6,20)(7,23)(8,24)(10,27)(11,30)(12,29)(13,28)(14,31)(16,32)(17,21)(19,22)(25,26);(1,5,6,3)(2,10)(4,18,20,15)(7,21,22,16)(8,13)(9,27)(11,25)(12,14)(17,19,32,23)(24,28)(26,30)(29,31)

Polycyclic presentation:

2^9|b^2=e,d^2=gh,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,f^b=fi,e^c=ei,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, 321227);