#383985

order	512
nilpotency class	3
isoclinism	isoclinic class of #31418
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2688
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14368
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

815 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,f^c=fi,g^c=gi,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, 291181);