#470943

order	512
nilpotency class	3
isoclinism	isoclinic class of #459756
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2761
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16719
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

685 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 378139);