#321501

order	512
nilpotency class	3
isoclinism	isoclinic class of #298795
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

789 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)(5,6,10,12,8,13,9,14)(7,15,18,20,16,11,17,19)(21,22)(23,30)(24,33)(25,29)(26,28)(27,32)(31,35)(34,36);(1,2,4,3)(5,7,8,16)(6,11,13,15)(9,17,10,18)(12,19,14,20)(21,23,24,34,25,32,26,31)(22,27,33,35,29,30,28,36);(6,12)(9,10)(11,19)(13,14)(15,20)(17,18)(21,23,25,32)(22,27,29,30)(24,31,26,34)(28,35,33,36);(1,3,4,2)(6,13)(11,15)(12,14)(19,20)(21,24,25,26)(22,28,29,33)(23,31,32,34)(27,35,30,36)

Polycyclic presentation:

2^9|a^2=fhi,b^2=eg,c^2=i,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=di,f^c=fh,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 228697);