#239862

order	512
nilpotency class	3
isoclinism	isoclinic class of #234079
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

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

978 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,g^a=gi,c^b=cg,d^b=dhi,f^b=fi,g^b=gi,d^c=di,f^c=fh,g^c=gi

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, 147058);