#414155

order	512
nilpotency class	3
isoclinism	isoclinic class of #395166
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

1028 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ei,c^2=e,d^2=gh,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,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, 321351);