#451646

order	512
nilpotency class	3
isoclinism	isoclinic class of #418871
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2697
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16006
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

754 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:

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

Polycyclic presentation:

2^9|a^2=efgh,b^2=ef,c^2=ef,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,h^b=hi,e^c=ei,f^c=fi,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, 358842);