#414143

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	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1064 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=ei,c^2=e,d^2=gh,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,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, 321339);