#288022

order	512
nilpotency class	3
isoclinism	isoclinic class of #282171
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#459	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

685 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)(5,6,9,15,10,13,8,14)(7,11,17,12,18,19,16,20)(21,22,24,28,25,29,32,27)(23,26,31,33,30,35,36,34);(1,2,4,3)(5,7,8,16,10,18,9,17)(6,11,14,20,13,19,15,12)(21,23,24,31,25,30,32,36)(22,26,28,33,29,35,27,34);(5,7,10,18)(6,12,13,20)(8,17,9,16)(11,14,19,15)(22,27)(23,30)(24,32)(26,33)(28,29)(34,35);(1,3,4,2)(6,13)(11,19)(12,20)(14,15)(21,24,25,32)(22,27,29,28)(23,31,30,36)(26,34,35,33)

Polycyclic presentation:

2^9|a^2=fhi,b^2=ef,c^2=hi,d^2=ei,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=deh,c^b=cg,g^b=gi,d^c=di,f^c=fh,g^c=ghi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 195218);