#437370

order	512
nilpotency class	3
isoclinism	isoclinic class of #429607
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2595
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15594
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

789 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:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=e,c^2=ef,d^2=ef,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^c=ei,f^c=fi,g^c=gi,g^d=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, 344566);