#423833

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,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, 331029);