#455947

order	512
nilpotency class	3
isoclinism	isoclinic class of #455828
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2631
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1175 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

(6,12)(7,17)(8,22)(9,10)(13,18)(14,29)(19,34)(20,21)(23,24)(25,28)(26,31)(27,30);(1,2)(3,4)(5,6,9,16,11,12,10,15)(7,18,17,25,21,28,20,13)(8,14,23,29,24,26,22,31)(19,33,32,30,34,36,35,27);(2,4)(5,7)(6,13,12,25)(8,19)(9,20)(10,17)(11,21)(14,30,26,27)(15,18,16,28)(22,32)(23,35)(24,34)(29,33,31,36);(5,8,11,24)(6,14,12,26)(7,19,21,34)(9,23,10,22)(13,27,25,30)(15,31,16,29)(17,32,20,35)(18,33,28,36)

Polycyclic presentation:

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

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, 363143);