#451049

order	512
nilpotency class	3
isoclinism	isoclinic class of #421163
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2420
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

1292 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 358245);