#206032

order	512
nilpotency class	3
isoclinism	isoclinic class of #205751
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#390
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

935 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,6,10,13)(4,8)(5,7)(9,14,11,16)(12,15)(17,18,22,27,23,28,21,26)(19,29,31,39,33,42,32,24)(20,34,36,41,37,40,35,25)(30,44,45,48,47,38,46,43);(1,3)(2,6,7,13)(4,11)(5,10)(8,16,15,14)(9,12)(17,19)(18,24,28,39)(20,30)(21,32)(22,31)(23,33)(25,38,41,44)(26,42,27,29)(34,43,40,48)(35,46)(36,45)(37,47);(2,7)(6,13)(8,15)(14,16)(17,20,21,35,23,37,22,36)(18,25,27,34,28,41,26,40)(19,30,32,46,33,47,31,45)(24,38,29,43,39,44,42,48);(1,4,5,12)(2,8,7,15)(3,9,10,11)(6,14,13,16)(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,35,37,36)(24,29,39,42)(25,40,41,34)(30,45,47,46)(38,43,44,48)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 113228);