#427397

order	512
nilpotency class	3
isoclinism	isoclinic class of #418498
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2673
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15369
prime radical	#23434	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1226 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=e,d^2=f,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,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, 334593);