#188054

order	512
nilpotency class	3
isoclinism	isoclinic class of #188049
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂²×#538	C₂
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

3010 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)(3,11,13,7)(4,15,17,8)(5,9)(6,10)(12,25,27,19)(14,24,29,20)(16,31,32,21)(18,22)(23,26,30,28);(1,3,6,14)(2,7,10,20)(4,12,17,28)(5,13,18,29)(8,19,15,30)(9,11,22,24)(16,26,32,27)(21,23,31,25);(1,4)(2,8)(3,12)(5,16)(6,17)(7,19)(9,21)(10,15)(11,23)(13,26)(14,28)(18,32)(20,30)(22,31)(24,25)(27,29);(4,16)(8,21)(12,26)(15,31)(17,32)(19,23)(25,30)(27,28)

Polycyclic presentation:

2^9|a^2=g,b^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^b=gh,d^c=dh,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 95250);