#456897

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

1171 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=ef,c^2=e,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,d^c=de,f^c=fi,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, 364093);