#455956

order	512
nilpotency class	3
isoclinism	isoclinic class of #455956
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16272
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23439	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1142 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)(3,14)(4,9)(5,13)(6,12)(7,11)(8,17)(10,21)(15,29)(16,23)(18,28)(19,24)(20,26)(22,27)(25,30)(31,32);(1,3,7,19,6,18,5,17)(2,8,13,28,12,24,11,14)(4,16,20,30,22,31,21,15)(9,25,10,23,27,29,26,32);(1,4,5,20,6,22,7,21)(2,9,11,10,12,27,13,26)(3,15,17,16,18,30,19,31)(8,23,14,29,24,32,28,25);(1,4)(2,10)(3,16)(5,21)(6,22)(7,20)(8,23)(9,11)(12,26)(13,27)(14,25)(15,17)(18,31)(19,30)(24,32)(28,29)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 363152);