#390342

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	#14569
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

750 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:

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

Polycyclic presentation:

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

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, 297538);