#288003

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

704 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)(5,6,8,13,9,12,17,11)(7,10,15,19,16,18,14,20)(21,22,26,28,25,30,24,29)(23,27,33,35,31,36,32,34);(1,2,4,3)(5,7,8,15,9,16,17,14)(6,10,13,19,12,18,11,20)(21,23,24,32,25,31,26,33)(22,27,29,34,30,36,28,35);(6,11)(7,14)(8,17)(10,18)(12,13)(15,16)(22,28)(23,31)(24,26)(27,34)(29,30)(35,36);(1,3,4,2)(6,12)(10,18)(11,13)(19,20)(21,24,25,26)(22,28,30,29)(23,32,31,33)(27,35,36,34)

Polycyclic presentation:

2^9|a^2=fi,b^2=ef,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, 195199);