#202101

order	512
nilpotency class	3
isoclinism	isoclinic class of #201948
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3444	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

984 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=i,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fhi,f^b=fi,e^c=ei,f^c=fh,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, 109297);