#459301

order	512
nilpotency class	3
isoclinism	isoclinic class of #455641
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2826
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

1127 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:

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

Polycyclic presentation:

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

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, 366497);