#391673

order	512
nilpotency class	3
isoclinism	isoclinic class of #32870
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2826
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14653
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

654 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)(3,12,16,13)(4,6)(5,9)(7,11)(8,15,24,22)(10,18)(14,28,32,29)(17,20)(19,26)(21,27)(23,31,25,30);(1,3)(2,8)(4,13)(5,14)(6,22)(7,16)(9,25)(10,12)(11,24)(15,18)(17,28)(19,31)(20,30)(21,32)(23,27)(26,29);(1,4,6,11,7,10,18,2)(3,13,15,8,16,12,22,24)(5,9,19,26,21,27,20,17)(14,25,30,28,32,23,31,29);(1,5,18,19,7,21,6,20)(2,9,10,26,11,27,4,17)(3,14,22,30,16,32,15,31)(8,23,13,29,24,25,12,28)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 298869);