#123479

order	512
nilpotency class	3
isoclinism	isoclinic class of #1828
automorphism	?
homology	Abelianization (H₁) = C₄³ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#381
commutator	C₂³	C₄³
Frattini, cosocle	C₂⁴×C₄	C₂³
socle	C₂²	C₂²⋊#95
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁

1730 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 30675);