#420726

order	512
nilpotency class	3
isoclinism	isoclinic class of #417464
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1235 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=e,c^2=ei,d^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=dei,f^c=fi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 327922);