#369259

order	512
nilpotency class	3
isoclinism	isoclinic class of #24468
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

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

1356 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=efg,d^2=ei,g^2=i,b^a=be,c^a=cf,d^a=de,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

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 276455);