#416981

order	512
nilpotency class	3
isoclinism	isoclinic class of #394801
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

876 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=eh,c^2=e,d^2=hi,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,f^b=fi,g^b=gi,d^c=di,f^c=fi,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, 324177);