#123849

order	512
nilpotency class	3
isoclinism	isoclinic class of #2048
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#394
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
socle	C₂²	#1442
prime radical	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁

1434 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=d,e^2=i,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fi,c^b=cf,e^b=ei,f^b=fi,g^b=gi,f^c=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 31045);