#261262

order	512
nilpotency class	3
isoclinism	isoclinic class of #249259
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	C₂³×C₄	C₂⁴
prime radical	C₂×#538	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1024 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)(5,6,9,13)(7,15,17,11)(8,14,10,12)(16,20,18,19)(21,22)(23,26)(24,27)(25,28)(29,34)(30,33)(31,35)(32,36);(1,2,4,3)(5,7)(6,11)(8,18)(9,17)(10,16)(12,19)(13,15)(14,20)(21,23)(22,26)(24,29)(25,30)(27,34)(28,33)(31,36)(32,35);(6,12)(8,10)(11,19)(13,14)(15,20)(16,18)(21,24,25,31)(22,27,28,35)(23,29,30,36)(26,32,33,34);(1,3,4,2)(5,8,9,10)(6,12,13,14)(7,16,17,18)(11,20,15,19)(24,31)(27,35)(29,36)(32,34)

Polycyclic presentation:

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

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, 168458);