#322564

order	512
nilpotency class	3
isoclinism	isoclinic class of #298939
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

718 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=eg,c^2=h,d^2=ehi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=dh,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, 229760);