#453904

order	512
nilpotency class	3
isoclinism	isoclinic class of #429607
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2595
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16210
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

682 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)(3,13)(4,18)(5,9)(6,11)(7,12)(8,15)(10,20)(14,23)(16,25)(17,24)(19,32)(21,26)(22,28)(27,29)(30,31);(1,3)(2,8,12,25)(4,19,21,29)(5,15)(6,16)(7,17)(9,24,11,13)(10,27)(14,20,30,22)(18,31)(23,26)(28,32);(2,9,12,11)(8,13,25,24)(10,26,28,18)(23,27,31,32);(1,4,7,21)(2,10,12,28)(3,14)(5,20,6,22)(8,23)(9,26,11,18)(13,27)(15,19)(16,29)(17,30)(24,32)(25,31)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 361100);