#249186

order	512
nilpotency class	3
isoclinism	isoclinic class of #249139
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

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

1261 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=hi,f^2=i,b^a=be,c^a=cf,d^a=deh,f^a=fi,c^b=cg,d^b=dh,f^b=fh,e^c=eh,f^c=fi,f^d=fh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 156382);