#297931

order	512
nilpotency class	3
isoclinism	isoclinic class of #271387
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2541	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1077 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)(7,14)(8,12)(9,18)(10,13)(11,15)(16,19)(17,20)(21,22)(23,27)(24,32)(25,34)(26,29)(28,33)(30,36)(31,35);(1,2,4,3)(21,23,26,31)(22,27,29,35)(24,33,30,25)(28,36,34,32);(5,7,9,15,10,16,8,17)(6,11,12,14,13,20,18,19)(21,24,25,23,26,30,33,31)(22,27,28,32,29,35,34,36);(1,3,4,2)(5,8,10,9)(6,12,13,18)(7,15,16,17)(11,19,20,14)

Polycyclic presentation:

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

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, 205127);