#423831

order	512
nilpotency class	3
isoclinism	isoclinic class of #418518
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
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

1293 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:

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

Polycyclic presentation:

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

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, 331027);