#437996

order	512
nilpotency class	3
isoclinism	isoclinic class of #417647
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

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

1153 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:

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

Polycyclic presentation:

2^9|b^2=eg,c^2=i,d^2=ei,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=de,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, 345192);