#146476

order	512
nilpotency class	4
isoclinism	isoclinic class of #12705
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
commutator	C₂⁴	C₂×C₄²
Frattini, cosocle	C₂³×D₄	C₂³
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1778 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)(7,15)(8,12)(9,21,23,13)(10,14)(11,17,25,35)(16,31)(18,30)(19,28,22,26)(20,27)(24,32,36,34)(29,33);(1,2,3,4)(5,7)(6,11)(8,16)(9,17)(10,18)(12,24)(13,15)(14,25)(19,32)(20,33)(21,30)(22,34)(23,35)(26,31)(27,36)(28,29);(5,8,10,20)(6,12,14,27)(7,16,18,33)(9,22,23,19)(11,24,25,36)(13,28,21,26)(15,29,30,31)(17,34,35,32)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53672);