#348291

order	512
nilpotency class	3
isoclinism	isoclinic class of #346898
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	C₂×#496	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

918 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:

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

Polycyclic presentation:

2^9|b^2=eh,c^2=i,d^2=hi,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=ghi,c^b=ch,f^c=fh,g^c=gi,f^d=fi,g^d=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, 255487);