#441591

order	512
nilpotency class	3
isoclinism	isoclinic class of #427311
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2471
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15687
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

920 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fg,c^2=f,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,e^c=ei,e^d=ei,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, 348787);