#441434

order	512
nilpotency class	3
isoclinism	isoclinic class of #427666
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2527
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15680
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

772 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:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=g,c^2=ef,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=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, 348630);