#465838

order	512
nilpotency class	3
isoclinism	isoclinic class of #455572
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
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

1043 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:

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

Polycyclic presentation:

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

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, 373034);