#238607

order	512
nilpotency class	3
isoclinism	isoclinic class of #233734
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	#37546	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1029 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 145803);