#434830

order	512
nilpotency class	3
isoclinism	isoclinic class of #434830
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15508
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

971 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 342026);