#448810

order	512
nilpotency class	3
isoclinism	isoclinic class of #448810
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15921
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

964 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:

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

Polycyclic presentation:

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

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, 356006);