#212007

order	512
nilpotency class	3
isoclinism	isoclinic class of #19456
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₄×#121
prime radical	C₂×#41	#41
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

528 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=eh,d^2=gh,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=ch,d^b=dhi,f^c=fi

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, 119203);