#123496

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

1766 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,f^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.

Relationships

maximal subgroup of	#72303562
maximal normal subgroup of	#72303562
maximal quotient of	#72303586

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 30692);