#123497

order	512
nilpotency class	3
isoclinism	isoclinic class of #1913
automorphism	?
homology	Abelianization (H₁) = C₄³ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
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

1734 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:

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

Polycyclic presentation:

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

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, 30693);