#202102

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	#60962	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1054 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)(3,10)(4,7)(9,14)(11,12)(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);(1,2,7,14,5,9,4,6)(3,8,10,15,12,16,11,13)(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)(2,8)(4,11)(5,12)(6,13)(7,10)(9,16)(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=egh,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, 109298);