#477003

order	512
nilpotency class	3
isoclinism	isoclinic class of #476613
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

1048 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 384199);