#368552

order	512
nilpotency class	3
isoclinism	isoclinic class of #368551
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14020
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1187 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)(3,4,14,18)(5,9,19,25)(6,11)(7,10)(8,17,21,26)(12,15)(13,16)(20,24,22,27)(23,32)(28,31)(29,30);(1,3,12,8,6,14,7,21)(2,4,15,17,11,18,10,26)(5,13,20,23,19,28,22,29)(9,16,24,32,25,31,27,30);(1,4)(2,8,11,21)(3,10,14,15)(5,16)(6,18)(7,17)(9,23,25,29)(12,26)(13,27,28,24)(19,31)(20,30)(22,32);(1,5)(2,9)(3,13)(4,16)(6,19)(7,22)(8,23)(10,27)(11,25)(12,20)(14,28)(15,24)(17,32)(18,31)(21,29)(26,30)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 275748);